Can't really call this an "urgent" care issue, as it's been going on for almost a year. I've tried treating it with tea tree oil and antibiotics, and I'm pretty sure it's not rot - but not sure. Is it growth? I thought at one point he was getting too much UVB, but after adjusting the amount of time and distance from the bulb, he still has the issue. When I scratch his shell with my fingernails (which he loves) I get little tiny pieces coming off under my nails. Are the white parts disease, molting or something else?
I suggest to do this overnight so your turtle can rest during this time. Dry dock for about minutes to make sure the shell is dry, then apply a very very thin layer to the affected areas. In the morning, wipe any residue left and put him back into the tank. The cream is a little greasy, so your turtle might be a little slippery. The cream should work quickly, if it does not then look into alternatives.