[Rusty]

dry mop (also called a dust mop)= mopa, recogepolvos
(Officially, there's no verb form in English, but we can make one up as we go because English allows this.)
to use a dry mop = pasar la mopa/el recogepolvos

I don't think a dust mop is dampened.
A regular mop can be slightly dampened and used, however.
(wet) mop = mopa, fregadora

frotar = rub (scrub, if done vigorously)
fregar = scrub (wash with a brush, towel, rag)
trapo = rag (a paño is a cloth used for cleaning, as well)


limpiar (el piso, etc., must have an object) con un trapo húmedo.

más húmedo que seco