I did my stint in the kitchen yesterday.

a period of time spent at a particular activity

Not much difference in meaning. Usage in normal speech might be slightly different. 'Spend some time' implies less time than 'spend time.' "Whenever I visit Tokyo I like to spend some time at my favorite restaurant." "I like to spend time with my family." The phrase 'some time' indicates an unspecified amount of time, but not a lot of time. The phrase 'spend time' does not actually indicate any amount of time. This leaves it open to being 'a lot' of time. If you said, "Whenever I visit Tokyo I like to spend time at my favorite restaurant." That would still be normal and natural but might imply you spend more time there than just 'some.' "I like to spend some time with my family." It sounds like maybe you like some time with your family and some time without them.

For me as a native speaker with imperfect grammar, I suggest the following: Both versions are correct. The first is a general statement that you regularly write for many hours (I spend hours writing [something]). The second version is in the past tense (I spent hours writing [something]). Therefore it has already finished. Finally, I would use the gerund (-ing) rather than the infinitive (to write) in both of these sentences unless I were to say something a bit different e.g. "It took me hours to write that" but.this is a very different sentence structure. I don't fully understand the grammar behind it, perhaps your teacher can help. Otherwise you might just be able to learn it as a phrase and work from there. Therefore I would actually say something like: I spend hours writing my English homework. (A regular activity) I spent hours writing my English homework. (A past activity) It took me hours to write my English homework. (Past tense again) There are many other options depending on the tense that you want to use. Good luck!