Is this a bite injury?
Or does it look like a hole with mucous in it and maybe something alive underneath?
I would suggest that this may be due to an infection under the skin causing an abscess that heals over on the top and breaks out again when the infection gets large enough.
The problem with penicillin its many organisms are not impacted by it anymore as its been around so long.
This site lists a penicillin dose for cats as you scroll down
as does this one
For home care you can use a warm/hot damp compress on the area to help it stay open and drain of infection. This will also stimulate blood circulation to help fight infection. Make sure to trim the fur back away from the wound area to help keep bacteria out of the injured area. You can clean the cuts out with Betadine iodine solution, chlorhexiderm solution, sterile saline, or 3% hydrogen peroxide. Then apply Neosporin or Triple Antibiotic ointment to the wound or any scabbed over areas. Keep the cat from licking the ointment off. It won't hurt the cat if it gets licked but it does better on the wound than on the tongue. A bit of hairball remedy on the roof of the mouth can keep the tongue busy while ointment soaks in.I'd do this 2-4 times a day and if you see no improvement or the infection seems to spread you really should consider seeing a vet if you are going to keep the cat.
Its always a good idea to consult your vet before trying any home care.Hope this helps you!