No brendon. it will kill the moss. do not even use fertilizer either.
best thing to do is to take the dish outside into the
fresh air and let it dry back some. The moss should
survive but I really would need to know what kind of moss it is and how you are growing it. Perhaps not enought
light and too much water or no drainage???

After I posted this question I decided to try my own way to get rid of the mold and retain moisture for the moss.

I took an appropriately sized glass panel, cleaned the glass with a gentle antibacterial hand cleanser, and put it on the dish. After almost a week there is no sign of the cotton-like mold and the moisture level has improved.

As to your questions:

I am not sure what species of moss I am growing, I don't even have a common name to go on. I found it growing in the shade of an old building and I live in north-central Kansas, if that helps at all.

I have simulated the moss's natural environment by giving it indirect sunlight throughout most of the day.

As for water I think I water it too much, I think it is because when I think of moss I think of sphagnum and lots of water so I tend to over water my moss. The moss is in terracotta so a small amount of water is released through the clay material.

I will try to upload a picture this evening.