I seriously doubt it is actual mould; most coinage metals are pretty good fungicides. Mould certainly cannot attack a coin directly. From the sounds of it, it's more likely to be PVC damage. Acetone will get rid of the PVC goo, but not the hazing - that's caused by corrosion, and no amount of washing in water, acetone or anything else will get rid of it.
But as a general rule, yes, you can simply "wash off" dirt - literal, mud-from-the-river dirt - in water. Preferably distilled/deionized water rather than ordinary tap water, especially for copper coins.
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Does a grading company ever clean a coin in any way, before grading it?
Both the major TPGs have conservation services; if you think a coin needs cleaning/conserving before grading, use their service. If you don't use their service and send them in a dirty coin, they won't clean it themselves. I think the most they will ordinarily do is give it a blast of canned air to remove dust.
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