Quietly feel offended by award (7)
I believe the answer is:
present
'award' is the definition.
(I know that award can be written as present)
'quietly feel offended by' is the wordplay.
'quietly' becomes 'p' (p for piano and quiet).
'feel offended by' becomes 'resent' (I've seen this in another clue).
'p'+'resent'='PRESENT'
(Other definitions for present that I've seen before include "Repents (anag.)" , "Exhibit" , "Gift; time now; here" , "It is given" , "host" .)