Broadcast of maximum annoyance (5)
I believe the answer is:
pique
'annoyance' is the definition.
(pique is a kind of annoyance)
'broadcast of maximum' is the wordplay.
'broadcast of' indicates a 'sounds like' (homophone) clue (I've seen 'broadcast' mean this (how the word might sound on the radio or TV)).
'maximum' becomes 'peak' (similar in meaning).
'peak' sounds like 'PIQUE'.
(Other definitions for pique that I've seen before include "Arouse" , "score at cards" , "Provoke" , "Wound (pride); resentment" , "Resentful feeling" .)