Could be feline acne or an allergy.
My kitty has the same problem. At first the vet diagnosed it as kitty acne. She would scratch until it bleed and it just seemed to last a long time w/o healing. She was put on steroids. This was the cycle for many years. It would come and go with frequency.
Then we moved to another state and we had found a kitty dermatologist at a emergency vet clinic. (really!) Turns out she has allergies. She was allergic to the perfume in the kitty litter! At least that is what we had determined after changing the cat litter to unscented. I still believe she has another allergy because she will "breakout" every so often - but it is more like once every year or two. The kitty dermie prescribed periactin and she takes that for it ONLY when it acts up. And then it is only half a pill twice a day for 7-10 days. I crush half the pill and mix it with her food since it has such a bitter taste - sometimes a SMALL amount of tuna as a treat. It is inexpensive too - only about $5 per occurrence.
Check with your VET BEFORE YOU PUT NEOSPORIN OR ANYTHING A HUMAN WOULD USE ON YOUR CAT! It may not be good for them or even poisonous. They have a different system then us. My vet told me peroxide is OK to use. Remember, they only know you put something sticky on them and they will lick it off and go into their system.
I live in NJ if you need to bring your kitty to a dermie I can recommend the place. My email is firstname.lastname@example.org
Pls use the same subject line otherwise I'll delete it. Good Luck!