Game flirt getting run in (7)
I believe the answer is:
croquet
'game' is the definition.
(croquet is a kind of game)
'flirt getting run in' is the wordplay.
'flirt' becomes 'coquet' ('coquet' can be a synonym of 'flirt').
'getting run' becomes 'r' (I can't justify this - if you can you should give a lot more credence to this answer).
'in' means one lot of letters goes inside another.
'coquet' enclosing 'r' is 'CROQUET'.
(Other definitions for croquet that I've seen before include "Game with wooden balls, mallets and hoops" , "Lawn game played with hoops and mallets" , "Game with balls and mallet" , "Lawn sport" , "Game in which a wooden ball is hit through hoops" .)