This is how I cleaned up and removed mildew that grew on a wood floor.
I left a mattress directly on a hardwood floor, and some mold grew underneath. There is no insulation in the crawlspace beneath the floor, so the temperature differential between warm bodies sleeping on the mattress and cold air under the floor resulted in condensation between the mattress and the floor, which is a perfect place for mold to grow.
Mildew can also grow under potted plants and carpet, against uninsulated walls, in damp basements, if you have a water leak, or even under wet laundry left sitting too long.
Step 1: Try Wiping Up the Mold
If the hardwood floor is properly sealed, you can just wipe up the mildew with vinegar, soap and water, dilute bleach, or some other common cleaning solution. In our case, most of the mold cleaned up easily, but some had gotten into the grain of the wood. The grain tends to suck up anything liquid, including the polyurethane used to finish our wood floors, which left those areas imperfectly sealed and able to absorb moisture. In these areas, the mold was able to grow a few millimeters into the wood, making it impossible to wipe away.
The mildew growing on the intact polyurethaned surfaces wiped up so quickly that I didn't get a picture of it before we cleaned it up.
Step 2: Sand the Floor
Using 100 grit followed by 220 grit, I sanded all the areas with mold. Since I didn't want to aerosolize a bunch of live mildew spores, I periodically sprayed a dilute solution of bleach on the areas I was sanding.
In the images, you can see that I tried removing the mold with steel wool. That was unsuccessful, so I sanded. If you use steel wool, be sure to clean all steel fibers up as any left may rust.
Step 3: Remove Dust
Since the polyurethane should be applied when the wood is dry, I removed the sawdust by dry wiping with microfiber followed by vacuuming.
Step 4: Refinish Hardwood Floor
This was in an occupied bedroom and I wanted it to dry quickly between coats, and so I chose water-based low-odor polyurethane to refinish the sanded areas. I applied a coat by brush in the morning, and the VOCs and solvent smell were long gone by bedtime. I repeated this for several days, sanding lightly between coats as directed by the poly's instructions.
If you use a cheap hog-bristle brush as I am in the images, be sure to remove any loose hairs before you start applying poly so those hairs don't come out and get stuck in the poly.
The floor was originally finished with an oil-based polyurethane, so I didn't expect the refinished areas to match exactly. However, they're normally out of sight (why mold grew there in the first place), and my objective was to seal-out further moisture rather than perfect them cosmetically. If this is a more visible area, test your polyurethane and consider applying a light stain. If the appearance really matters, you should probably refinish the entire floor.
I made sure mold wouldn't be a problem again in the future by building a low-profile bed frame.