Keep putting on the Neosporin without pain on the wounds. You can also put on some hydrocortisone (anti-itch cream), and it may help with any irritation.
When they have eye irritation, they produce extra liquid from their tear ducts. It can get caked on their feathers near their eyes. I would take a damp cloth and wipe it off.
As long as you are making sure he eats food, particularly pellets, and is drinking, there is not a whole lot more that you can do other than give it time. You could do antibiotics, but that may or may not help. The terramycin is probably the main one to try. You can get it at TSC, chewy.com, or
https://allbirdproducts.com/products/terramycin-ointment
Right now it is just going to take time. It may take weeks before he feels himself again.