What is the Halacha of a Jewish married man engaging in physical relations with a Jewish divorced woman.
Let's assume the divorced Jewish woman had a proper, halachic Gett, so she is actually divorced from the perspective of halacha. (Otherwise this gets very dicey, very fast.)
Because Judaism allowed a man to have more than one wife, until a thousand years ago, technically a married man could go find another woman without breaking halacha, and thus the strictest halachic definition of "adultery" only concerns relations between a married woman and a man who's not her husband.
Still, it should go without saying that for a married man to cheat on his wife is despicable, wrong, a violation to fellow human beings, and a sin to God. (At the bare minimum, all non-marital relations are prohibited.) The technical term "adultery" is not used, however.
Presumably if the couple got divorced, the husband's infidelity would be considered when the assets are being divided.
Anyhow, back to the theoretical prohibition for the act itself -- we'd fall back on if both of them were single (and let's assume it's clear their intent is purely promiscuous.) If the woman is a Nida, that's prohibited and theoretically punishable by karet -- G-d takes it up with their soul. If not, then it's theoretically punishable by lashes (either at the rabbinic or Biblical level, depending on which medieval interpretation we follow).
Nonetheless, Sharon Galper Grossman describes the phenomenon you describe as happening occasionally in Israel in a recent journal article about HPV; in a subsequent letter to the editor, she acknowledged that maybe that wasn't the best thing to print.
If the woman later remarries, her kesubah will say exactly the same thing whether she'd had relations with someone or not, simply identifying her as a divorcee (divorcees can never marry a kohen anyhow, so that point is moot).
If she got pregnant from the union with the Jewish married man: if the father wasn't a kohen, then the child is a 100% kosher Jew with no special limitations whatsoever [because theoretically the parents could have married at the time]. If the father was a kohen, then the child was born of a forbidden union -- if it's a boy he is not a kohen, if it is a girl she may not marry a kohen.