Prudish Greek character backing church dignitary (7)
I believe the answer is:
primate
'church dignitary' is the definition.
'prudish greek character backing' is the wordplay.
'prudish' becomes 'prim' (synonyms).
'greek character' becomes 'eta' (letter from the Greek alphabet).
'backing' is a reversal indicator.
'eta' written backwards gives 'ate'.
'prim'+'ate'='PRIMATE'
(Other definitions for primate that I've seen before include "Man (or monkey)" , "bishop, perhaps" , "Metropolitan" , "Bishop; large mammal" , "Ape - cleric" .)