Awfully bitter adult in a cap (7)
I believe the answer is:
biretta
'a cap' is the definition.
(biretta is a kind of cap)
'awfully bitter adult' is the wordplay.
'awfully' indicates anagramming the letters.
'adult' becomes 'a'.
'bitter' is an anagram of 'birett'.
'birett'+'a'='BIRETTA'
'in' is the link.
(Other definitions for biretta that I've seen before include "Stiff clerical cap" , "Roman Catholic priest's cap" , "Square cap worn by Roman Catholic clergy" , "Battier (anag.)" , "Cleric takes this on himself" .)