Lover left out celebrity (4)
I believe the answer is:
fame
'celebrity' is the definition.
('fame' can be a synonym of 'celebrity')
'lover left out' is the wordplay.
'lover' becomes 'flame' (I've seen this in other clues).
'left' becomes 'l' (common abbreviation).
'out' is a deletion indicator.
'flame' with 'l' taken out is 'FAME'.
(Other definitions for fame that I've seen before include "''... is the spur that the clear spirit doth raise (Lycidas, John Milton)" , "Notoriety or celebrity" , "Prominence" , "Favourable public reputation" , "Repute" .)