What might be a good way to fix this damaged drywall(?)?
It looks like one layer of paper has separated from the other. The paper surrounding the tearout has loose flaps of paper around it too.

Remove all loose paper and give a tight skim coat or two of paste spackle, sand smooth, prime and paint.



You will be wise to get a 4" drywall knife to help in this. The little 1" knife that is considered a putty knife are good for nail holes and window glazing. Check out a YouTube tutorial on drywall paper tear repair