Sand down the top and refinish.

This was my first thought, but I'd guess that any staining (whether
mold or other discoloration) would penetrate at least 1/64". Sand
lightly so you don't go through the veneer! I'd try bleach, as that
should kill and remove anything fungal (mold/mildew). Can't hurt, and
even if it does, you're not out too much as the top is already
discolored. I wouldn't use a large volume of bleach or anything
water-based, as that could swell/disintegrate the MDF.
I'd plan on replacing or refacing the top (1/4" red oak ply isn't too
expensive, and could be glued right on top of the old surface, and then
maybe the whole top could be edge-banded?). That way, if none of the
stain removal or sanding procedures work, you won't be too
disappointed, and you can just move on to the new surface.
Good luck,
Andy