There has been a high-profile alimony case in India recently (which was also reported on the BBC website), where two senior Supreme Court judges referred to a man's mistress as a keep. A senior female lawyer strongly objected to the term keep, saying she found it personally offensive, and regarded the judge's words as being in extremely poor taste (see this Times of India article).
The specific 'mistress' meaning of the noun keep is not in the OED; the nearest definition is one that Edmund Spenser used, namely "that which is kept; a charge". However, the phrase 'kept woman' is not uncommon, and is often used humorously these days, as, indeed, is kept man. The OED also has the definition "one who keeps a mistress" at the headword keeper, which it describes as obsolete and it has the entry keeperess, one of whose meanings is "a woman who keeps a man". Similarly, at 'keeping' one of the senses is "the maintaining of a mistress or lover".
There are dozens of words in the OED which are synonyms of the word 'mistress', or which have been at some time - with the 17th century being the heyday for such vocabulary. Here are some: buttered bun, convenient, doxy, fancy-woman, gimmer, jilt, jug, kittock, leveret, liberty-wife, lie-by, Lindabrides, minion, minx, mule, natural, night piece, nobsy, peculiar, pinnace, pop, poplolly, pug, pure, sooterkin, special, tackle, underput and wanton.