How large is the gravel? Larger gravel shifts when corys are feeding and can injure the barbels of some species. I try and keep my longer nosed corys in #0 - #2 quartz gravel... thats significantly smaller than what is normally available in the pre-washed estes etc. gravel bags. I had identical problems to you when I started out in the hobby and used large gravels.
You might also be seeing the results of stress from water quality... have you tested ammonia, nitrite, and nitrate? Hows the temperature? Are you doing frequent water changes? Its always easier for fish like corys to go down this road if the water parameters are off.
For treatment, I would keep it simple at the moment. If you find any of the chemistry is off, correct it. Probably do a large water change regardless and keep to a more frequent and regular schedule. Also, its always good to add 2 teaspoons of kosher or aquarium salt per gallon of aquarium water if you don't have any plants. The salt will help keep down infection and may help the cory regrow the damaged parts.
Unfortunately, times when I've seen cory cats develop serious infections around their mouthparts they just haven't been able to come back from it - they are small little creatures and it doesn't take much to overwhelm their little systems. Generally you will see redness, swelling, or fungus if the infection is out of control. If you wanted to add an antibiotic I would try Erythromycin... but salt is a simpler and safer choice if its only the one fish.
Hope some of this helped
-Sam
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