I can't think of a single word, except maybe hours. People talk about the hours they worked on something. "How many hours do you have in this week?"

I saw that one time-tracking site called time spent on a task "active time," and another called it "tasktime" (one word, but coined, I think). In an office setting, it could also be called "billable time" or "billable hours."

You ask someone else is they can spare the time for something.

Excuse me, could you please spare a few moments of your time?

If you refer to yourself

I can spare some time to talk to you

it makes you sound self-important or condescending.

In the context of your loved ones, again it makes it seem that they are unimportant, if you say you can spare some time to be with them. So

I like to spend time with my loved ones.

You can also ask someone to spend some time with you, if you want to be friendly:

I'd really like it if you could spend some time with me.

If you can spare some time, you make the time available.
If you spend time, you are using the time that you have.

To complicate it slightly further, you might have some spare time that you can spend doing something.