Green’s Dictionary of Slang

1. to trifle with, to fool around.

1889–90		Barrère & Leland Dict. of Sl., Jargon and Cant.
1920		R. Lardner Big Town 164: The Mrs. told Kate she was a hick to be monking round with a jockey.
1949		J.H. Burns Lucifer with a Book 180: He looked from his windows at the moonlight monking it up with the breast of the nunlike snow on the meadows.

2. to neck, to engage in sexual activity.

1960, 1975

Wentworth & Flexner DAS 343/1: monk [...] To neck. Some c1935 student use; never common.