Since dogs are constantly running, and on there feet, a cut on a pad is very difficult to heal. Peroxide for cleaning is fine. If you take your dog to the vet, you'll have a hundred dollar bill and nothing can be done. Buy a tube of super glue and put three coats of super glue over the cut. Do you have a muzzle? Before you put it on, you need to find a way to prevent the dog from licking it off. It only takes a minute for the super glue to dry totally, and this will give your dog a protective cover over the pad that will last awhile and will allow it to heal completely.
