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 bite. 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. Distract the cat for a few minutes perhaps by putting a little bit of butter or canned cat food on the top of a paw to keep it busy licking there.I'd do this 2-4 times a day and if you see no improvement or the infection seems to spread you should see your vet for a check up and possibly oral antibiotics or other care.
If this is Sudocrem you are using or another product check for zinc as an ingredient. If Zinc is one of the main ingredients you might want to use something else as zinc can be toxic if ingested. You can read about zinc and other heavy metals here
Hope this helps you!