Part of church with luck found on left (7)
I believe the answer is:
chancel
'part of church' is the definition.
('chancel' is part of 'church')
'luck found on left' is the wordplay.
'luck' becomes 'chance' ('chance' can be a synonym of 'luck').
'found on' is a charade indicator (letters next to each other).
'left' becomes 'l' (common abbreviation).
'chance'+'l'='CHANCEL'
'with' acts as a link.
(Other definitions for chancel that I've seen before include "Clergy's area of the church" , "church, partly" , "where sanctuary may be found" , "choir and clergy's area" , "Part of church near altar" .)