Broadcast regretted being impolite (4)
I believe the answer is:
rude
'impolite' is the definition.
(similar in meaning)
'broadcast regretted' is the wordplay.
'broadcast' shows a homophone (sound like) (how the word might sound on the radio or TV).
'regretted' becomes 'rued' ('rue' can be a synonym of 'regret').
'rued' is a homophone of 'RUDE'.
'being' acts as a link.
(Other definitions for rude that I've seen before include "Insolent" , "Coarse, unpolished" , "Impolite - obscene" , "Base" , "Impolite; unpolished" .)