Put away tailing puritanical bishop (7)
I believe the answer is:
primate
'bishop' is the definition.
(I've seen this before)
'put away tailing puritanical' is the wordplay.
'put away' becomes 'ate' (put away can mean to eat).
'tailing' means one lot of letters go next to another (to tail can mean to go after or follow).
'puritanical' becomes 'prim' (synonyms).
'ate' put after 'prim' is 'PRIMATE'.
(Other definitions for primate that I've seen before include "Ape or churchman" , "Higher mammal" , "Capuchin could be such" , "Chief priest" , "Highest-ranking bishop" .)