Exaggerate having left party (6)
I believe the answer is:
overdo
'exaggerate' is the definition.
('overdo' can be a synonym of 'exaggerate')
'left party' is the wordplay.
'left' becomes 'over' (I've seen this before).
'party' becomes 'do' (do is a kind of party).
'over'+'do'='OVERDO'
'having' acts as a link.
(Other definitions for overdo that I've seen before include "Exaggerate; use too much of" , "Put too much effort into" , "Use to excess" , "Exhaust" , "Perform excessively" .)