The sink leaked and there is black mold on the adjoining section of
The area under the sink. Even breathing near the mold is irritating to my throat.

It sounds like you are sensitive to mold, and without proper breathing protection, just stay away. There are products out there that "KILL" mold; but first you have to stop the problem that caused mold to grow in the first place. You will need a HEPA vacuum. Mold spores are between 1 and 20 microns in size, so a regular machine will spread them all over the place really fast! You will need protective clothing, gloves, goggles, etc.. Borax and water (about 1 cup per gallon) hydrogen peroxide (right off the shelf) can each be used to wet down and kill the stuff. Wipe off the residue after letting the cleaning agent work for at least ten minutes. Vacuum the area when dry. Then apply a stain blocking sealer like white pigmented shellac (original KILZ, or BIN). Borax can be found in the laundry section of your local store, hydrogen peroxide in first aid section. Either will work. Both are inexpensive. The vacuum $400, breathing respirator and proper cartridges $40, rubber gloves (good ones) $5... If you can't protect yourself properly, and do it right, get professional help. Remember porus surfaces trap mold. Cleaning only the surface it will grow back. Helpful?