Hi Ladies;
Whenever I clean my sinks I get a wad of paper towels and wet them with whatever I'm using. Put your hand with the towels down the hole and then bring the towels upward underneath the black gizmo and wipe real good. Bring your hand out, turn the towels to a clean area and do it again.
This method really gets a lot of yucky gunk out of there. Dispose of the towels and clean your hands scrupulously.
Next time I intend to cover my hand with a plastic bag first as someone mentioned here.