I think you have two options here.

Either you repair that hole by filling it with epoxy resin or another suitable filler. After that you may have to drill it again to make sure it has the right diameter and position.

Or you drill a bigger hole and use an insert nut (see wikipedia). Then you also have to replace the special scew depicted by a suitable countersunk machine screw. Of course it depends on the thickness of your board, whether you can use an insert nut.

Answer from Ariser on Stack Exchange
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Amazon
amazon.com › Drawer-Doctor-Broken-Buckled-Minutes › dp › B077NF5W54
Amazon.com: DIY Doctor Drawer Repair Kit (6 Pack) – Fix Sagging Kitchen Drawers – Front Fixing Brackets & Bottom Support – Replacement for Damaged Units – Draw Strengthener Pack : Tools & Home Improvement
Buy DIY Doctor Drawer Repair Kit (6 Pack) – Fix Sagging Kitchen Drawers – Front Fixing Brackets & Bottom Support – Replacement for Damaged Units – Draw Strengthener Pack: Drawer Slides - Amazon.com ✓ FREE DELIVERY possible on eligible ...
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Amazon
amazon.com › drawer-repair-kit › s
Amazon.com: Drawer Repair Kit
FRMSAET Drawer Repair Kit - Used to Reinforce and Repair Wooden/MDF/Chipboard Drawers Cabinet Reinforcement Heavy Duty Steel Hardware Furniture Accessories - Includes Screws (2 Pack) · FRMSAET Drawer Repair Kit - Used to Reinforce and Repair Wooden/MDF/Chipboard Drawers Cabinet Reinforcement ...
Discussions

repair - How can I reattach sliding drawers which have damaged their screw holes? - Home Improvement Stack Exchange
We have a set of sliding kitchen drawers which came loose, probably due to overloading, damaging the holes which they screwed into, as in this image: The screws themselves look like this: (apologies if "screw" is the wrong word) We reordered the drawers to make use of the remaining undamaged screw holes, but they've come loose again, and now there are no undamaged screw holes to use. Is it possible to repair ... More on diy.stackexchange.com
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September 19, 2018
How do I fix a sagging drawer?
I’m sure there will be other, better suggestions than mine, but just in case I’ll still leave a suggestion. You can add a brace underneath the board. Fit that board back into its slot, cut a piece of wood that runs from front to back and fit it underneath the drawer. Pinning it in with a pinner would be easiest, nailing it in would be fine, screws would be good too. If you nail or screw it in you’ll want to predrill the holes so you don’t crack the wood. More on reddit.com
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33
88
May 5, 2022
How to repair wooden drawer slides on vintage dresser

The pieces on the insides of the dresser need to be replaced. If you have or know someone with a table saw, they are pretty easy to fabricate. Or take a sample to a woodshop, i can't imagine they would charge too much. Then you would remove the old and install the new pieces.

If the edges of the drawers are worn, sometimes I'll get 1/4" square rods from the hardware store, cut to size, glue and nail it to the worn bottom edges and use a sander or something to get it to the right shape.

More on reddit.com
🌐 r/furniturerepair
3
1
October 12, 2023
Kitchen drawer repair. Plastic white piece. Mid to late 1980's cabinet
I would also suggest taking a look at part 32-053 as an alternative if you feel part 32-078 won't be an appropriate fit. it is a complete kit that would replace the entire old track system. More on swisco.com
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1
November 23, 2022
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The Home Depot
homedepot.com › p › Prime-Line-24-in-Wood-Drawer-Track-Repair-Kit-R-7144 › 100162824
Prime-Line 24 in., Wood, Drawer Track Repair Kit R 7144 - The Home Depot
Prime-Line 24 in., Wood, Drawer Track Repair Kit R 7144
This kit is constructed of a wooden track, steel brackets and a plastic guide and back plate, together with all the parts necessary to repair one wooden drawer track system. Track restores smooth operation to track mounted drawers, and it can be used for top or bottom mount installations. This universal kit allows the 24 inch wooden track to be cut down to size to accommodate for shorter drawer applications. The white plastic saddles are used to replace worn or missing guide buttons, enabling the drawers to glide easily while at the same time preventing wear on the cabinet's face frame or draw
(4.2)
Price   $6.08
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Walmart
walmart.com › home improvement
Home Improvement Drawer Repair Kits for Easy Furniture Fixes
$1413current price $14.13110 Sets Furniture Chest Drawer Kit Dresser Bottom Repair Fixing Mending Wedges Bracket Pallet Furniture Support Brackets 90 Degree Drawer Plastic with Screws (White) for DIY
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Prime-Line Parts
primelineparts.com › drawer-track-repair-kit-metal-track-22-1-2-in-long-1-kit
Drawer Track Repair Kit, Metal Track, 22-1/2 in. Long (1 Kit) - Prime-Line Parts
Drawer Track Repair Kit, Metal Track, 22-1/2 in. Long (1 Kit)
This kit is constructed of steel and plastic components and it includes all the parts necessary to repair one wooden drawer track system. Track...
Price   $9.72
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eBay
ebay.co.uk › drawer repair kit
drawer repair kit products for sale | eBay UK
Buy drawer repair kit products and get the best deals at the lowest prices on eBay UK! Great Savings & Free Delivery / Collection on many items
Find elsewhere
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McCoy's
mccoys.com › shop › p › 0147462-052303-28 › prime-line-r7144-drawer-track-repair-kit-steel
R7144 Drawer Track Repair Kit, Steel
prime-line-r7144-drawer-track-repair-kit-steel
Complete kit designed to replace most wood track systems. May be used for top or bottom mount installations. Restores smooth operation to track mounted drawers. Track may be cut down to fit shorter drawer applications. Complete kit designed to replace.
Price   $11.00
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JML Hardware
jmlhardware.co.uk › home › kitchen repair plates › drawer repair plate
Drawer Repair Plate | JML Hardware
December 20, 2024 - Buy our Drawer Repair Plate from JML Hardware as over time they can become broken our repair plates is easy to install, In stock
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Reddit
reddit.com › r/furniturerepair › how to repair wooden drawer slides on vintage dresser
r/furniturerepair on Reddit: How to repair wooden drawer slides on vintage dresser
October 12, 2023 - Furniture and Architectural, also Repairs to Ceramic, Leather, Art, Metal, Marble or other related topics! ... The pieces on the insides of the dresser need to be replaced. If you have or know someone with a table saw, they are pretty easy to fabricate. Or take a sample to a woodshop, i can't imagine they would charge too much. Then you would remove the old and install the new pieces. If the edges of the drawers are worn, sometimes I'll get 1/4" square rods from the hardware store, cut to size, glue and nail it to the worn bottom edges and use a sander or something to get it to the right shape.
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Menards
menards.com › main › hardware › cabinet-hardware › cabinet-hardware-accessories › prime-line-reg-24-wood-drawer-track-repair-kit › r7144 › p-1444451502912-c-1526043700165.htm
Prime-Line® 24" Wood Drawer Track Repair Kit at Menards®
Prime-Line® 24" Wood Drawer Track Repair Kit
Prime-Line® Products has been supplying repair and replacement parts for windows and doors since 1978. This drawer track kit from Prime-Line® features a wooden track measuring 24 inches long. The track itself boasts a universal design that can fit most wood track systems, and it can be cut down in length for more shallow applications. This kit also includes a plastic guide, a back plate, two side saddles, and two steel angle brackets to support the front end of the track. This type of track kit can be used for top or bottom mount applications and includes all necessary fasteners for quick and
Price   $10.67
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Quora
quora.com › How-can-you-get-a-drawer-back-on-track-after-it-slides-out-wrong
How to get a drawer back on track after it slides out wrong - Quora
Answer (1 of 3): Most drawers run on metal slides that hold the drawer straight. One just needs to pull the drawer out and re-aim it level and straight into the cupboard slot.
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The Home Depot
homedepot.com › diy projects & ideas › home decor ideas › kitchen ideas & projects
How to Fix a Broken Drawer - The Home Depot
September 5, 2023 - Kitchen · 1 · Assess the Problem · 2 · Fix a Broken Drawer Front · 3 · How to Fix Drawer Inner Panels · 4 · Fix the Broken Sliders on a Door · 5 · Tools Needed to Fix a Broken Drawer · As a general rule, drawers seem to break down in three (3) basic ways: The front panel and/or hardware get loose or damaged. Gaps occur on a drawer's inside bottom panel. The drawer's sliders wear out. Before starting a drawer repair, empty the drawer and remove it for a thorough inspection.
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Tiny House Me
tinyhouseme.com › product › Kit-8-Piece-Set-To-Fix-Sagging-Broken-Drawer-Bottoms › 551315
8-Piece Set To Fix Sagging & Broken Drawer Bottoms ...
Bedroom and Space Requirements: A tiny home must have a room of at least 70 square feet, a functional bathroom with a closet and window, and bathrooms and kitchens with a minimum ceiling height of 6’4” to be up to the building codes.
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iFixit
ifixit.com › Answers › View › 835469 › How+to+repair+broken+dresser+drawer
How to repair broken dresser drawer - Burrow Sectional Furniture - iFixit
February 5, 2024 - @stacymauceri possible fix with some wood glue or some screws. Let us see what this looks like so we can give you some clear directions. Post some well focused question of your drawer front and how it currently is mounted, with your Question.
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Friedman's Home Improvement
friedmanshome.com › 24-in-wood-drawer-track-repair-kit-4376347
Prime-Line | 24 in. Wood Drawer Track Repair Kit | Friedman's Home Improvement
24 in. Wood Drawer Track Repair Kit
This kit is constructed of a wooden track, steel brackets and a plastic guide and back plate, together with all the parts necessary to repair one wooden drawer track system. Track restores smooth operation to track mounted drawers and it can be used for top or bottom mount installations. This universal kit allows the 24 in. wooden track to be cut down to size to accommodate for shorter drawer applications. The white plastic saddles are used to replace worn or missing guide buttons, enabling the drawers to glide easily while at the same time preventing wear on the cabinets face frame or drawer
Price   $5.63
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SWISCO
swisco.com › home › kitchen drawer repair. plastic white piece. mid to late 1980's cabinet
Kitchen drawer repair. Plastic white piece. Mid to late 1980's cabinet : SWISCO.com
November 23, 2022 - I would also suggest taking a look at part 32-053 as an alternative if you feel part 32-078 won't be an appropriate fit. it is a complete kit that would replace the entire old track system.
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Lowe's Pro Supply
lowesprosupply.com › Catalog › Hardware › Cabinet-And-Drawer-Hardware › Drawer-Guide-Repair
Drawer Guide Repair
11 Items in Drawer Guide Repair · Compare · Drawer Side-Mount Slide Sockets · 642165 · Number · - + Add To Cart · Item Added Add To List · Add To List · Select a list or enter a new wish list name · Item already exists · Select List · Create New List · - OR - Create New List · Please Sign In. Schedule email reminders to reorder this list · Cancel · Add to List · Compare · Drawer Track Kit Monorail 22-5/8" 642110 ·