After you get the mold controlled, consider spraying the spots, and really anything else, with sodium carbonate (a.k.a. washing soda or soda ash). Don't confuse that with sodium
bicarbonate a.k.a. baking soda. It's sold in hardware stores already mixed as a mold-killer called Concrobium. Basically it works mechanically, not chemically, by drying and crushing mold spores. Safe for lungs, kids, pets, etc.
Just an hour ago I had to tear out some moldly particle board from my mother-in-law's mobile home sink. I sprayed down pure bleach first, wiped off the surface mold and then doused it all really well with sodium carbonate.
Oh, I see they're now using baking soda in their formula as well.
Article on ConcrobiumHow to Make Mold Cleaner With Sodium CarbonateSodium Carbonate on Amazon