Offence, annoyance (7)
I believe the answer is:
umbrage
I believe this clue is a double definition.
'offence' is the first definition.
(I know that offence can be written as umbrage)
'annoyance' is the second definition.
(I've seen this in another clue)
(Other definitions for umbrage that I've seen before include "Take this to show offence" , "Offence that's usually taken" , "In high dudgeon" , "The sulks" , "Offence, annoyance" .)