Hang around after pub room’s backed Wild West fighter (10)
I believe the answer is:
gunslinger
'wild west fighter' is the definition.
'hang around after pub room's backed' is the wordplay.
'hang around' becomes 'linger' ('linger' can be a synonym of 'hang around'**).
'after' is a charade indicator (letters next to each other).
'pub room' becomes 'snug'.
'backed' says the letters should be written backwards.
'snug' back-to-front is 'guns'.
'linger' put after 'guns' is 'GUNSLINGER'.
(Other definitions for gunslinger that I've seen before include "Armed cowboy perhaps" , "One with a hand weapon" , "Armed bandit" , "shot by him?" , "Fighter" .)