A bishop's rotten apple is unfinished Cox (9)
I believe the answer is:
episcopal
'a bishop's' is the definition.
(I know this)
'rotten apple is unfinished cox' is the wordplay.
'rotten' is an anagram indicator.
'unfinished' means to remove the last letter.
'cox' with its final letter removed is 'co'.
'apple'+'is'+'co'='appleisco'
'appleisco' is an anagram of 'EPISCOPAL'.
(Other definitions for episcopal that I've seen before include "Pepsi-Cola (anag) - pertaining to a bishop" , "run by bishops" , "To do with bishop - maybe Pepsi-Cola" , "Relating to a bishop" , "controlled by bishop" .)