Bitterness caused by broadcast of Asterix, perhaps (4)
I believe the answer is:
gall
'bitterness' is the definition.
(synonyms)
'broadcast of asterix perhaps' is the wordplay.
'broadcast of' shows a homophone (sound like) (I've seen 'broadcast' mean this (how the word might sound on the radio or TV)).
'asterix perhaps' becomes 'Gaul' (Asterix is a Gaul in a series of comic books).
'gaul' sounds like 'GALL'.
'caused by' acts as a link.
(Other definitions for gall that I've seen before include "Bold impudent behaviour; abnormal growth on tree" , "Impudence, cheek" , "Impertinence" , "Bitterness of feeling or bodily bladder" , "Impudence; bitterness" .)