A bitter drunk — it may go to cardinal's head (7)
I believe the answer is:
biretta
'it may go to cardinal's head' is the definition.
(cap worn by clergy)
'a bitter drunk' is the wordplay.
'drunk' indicates an anagram.
'a'+'bitter'='abitter'
'abitter' anagrammed gives 'BIRETTA'.
(Other definitions for biretta that I've seen before include "Battier (anag.)" , "religious headwear" , "Square cap worn by Roman Catholic clergy" , "Cloth cap" , "Cleric takes this on himself" .)