Lying Liberal stops supporting women's sport (9)
I believe the answer is:
wrestling
'sport' is the definition.
(I know that wrestling is a type of sport)
'lying liberal stops supporting women's' is the wordplay.
'lying' becomes 'resting' ('rest' can be a synonym of 'lie').
'liberal' becomes 'l' (abbreviation).
'stops' means one lot of letters goes inside another (stopping can mean blocking a gap).
'supporting' means one lot of letters go next to another (in a down clue, some letters go below others).
'women' becomes 'w' (standard abbreviation).
'resting' going around 'l' is 'restling'.
'restling' put after 'w' is 'WRESTLING'.
(Other definitions for wrestling that I've seen before include "Hand-to-hand combat" , "See 8" , "Ring sport" , "Graeco-Roman and freestyle Olympic events" , "event" .)