I was surprised not to find this listed in the dictionary of slang/colloquialisms. It's frequently used to mean cheating on your partner, yes.
He's been playing away.
She's playing away from home.
It's based on the football term where a team plays a match in another city. If they play the match in their own city, at their own stadium it's a home game, or they play at home. If they travel for a match, they play away [from home].





