I'd find an exotics veterinarian in your area and see if you and your little one can get an appointment. I'm not familiar with how those injuries can look on an amphibian specifically but that looks like a spreading infection to me. Neosporin could cause the wound to close over the infection, trapping puss, etc in the body. Assuming it works on a frog the same as it would a person.

Definitely try to find a vet, you might save his life!