If ενα παιχνίδι is a toy or a game, it's also a "plaything." Plaything was marked wrong, and "a playing" was given as the correct version. In no dialect or regional style of any world english have I ever seen "a playing" as an accepted phrase. Please chime in and tell me if you would use that (Native American English speaker asking this).
So, sorry and many thanks for catching that. "Plaything" was either misspelled or automatically "corrected" to "playing" in the incubator. I've corrected it. And to answer your question..."No, I would never use it nor have I ever heard it in either AE of BE." As I said it was our typo.