Pole seen after girl quits (4)
I believe the answer is:
even
'quits' is the definition.
(as in 'call it quits')
'pole seen after girl' is the wordplay.
'pole' becomes 'n' (abbreviation for North, as in North Pole).
'seen after' means one lot of letters go next to another.
'girl' becomes 'eve' ().
'n' after 'eve' is 'EVEN'.
(Other definitions for even that I've seen before include "Flat - yet" , "Fair" , "50-50" , "time of day" , "Level as odds may be" .)