Keen on a bit of badminton (4)
I believe the answer is:
into
'keen on' is the definition.
(I've seen this before)
'a bit of badminton' is the wordplay.
'a bit of' indicates the answer is hidden within the clue (I've seen 'a bit' mean this (the answer is formed by a bit of the clue)).
'INTO' can be found hidden inside 'badminton'.
(Other definitions for into that I've seen before include "Taking an active and lively interest (in something)" , "digging" , "Liking" , "Fascinated by" , "To a position within" .)