Well, I'll display my ignorance and ask is the crud on the gate, in the eye and ear of Contantine and in front of the bust how this coin is supposed to be kept? (I naturally want to get it off.) Also, is the coin oiled? Is that where the shine comes from?
Honestly, I'd just leave it the way it is. If you were to clean it, practice (for about a year) on more expendable coins. I can't really tell from the pic, the glossiness might be a result of the way the hoard was processed. Or it could be natural hoard patina.
I suspect the areas in question are where heavier deposits were removed and the glossy areas are the more natural surface of the coin. Often coins can be cleaned but what is underneath the crud is no better looking than what was removed. The area infront of the bust especially looks porous like material such as seen in the ear was removed from that area. I would leave it alone. Time could even out the color of the tones but the pits are there for good.
