Run a piece rewritten by bon vivant (9)
I believe the answer is:
epicurean
'bon vivant' is the definition.
('epicurean' can be a synonym of 'bon vivant')
'run a piece rewritten' is the wordplay.
'rewritten' is an anagram indicator.
'run'+'a'+'piece'='runapiece'
'runapiece' anagrammed gives 'EPICUREAN'.
'by' is the link.
(Other definitions for epicurean that I've seen before include "Bon vivant" , "follower of luxury" , "one sensuous and Greek" , "Foodie's" , "pleasure-seeker" .)