Regretted being impolite, we hear (4)
I believe the answer is:
rued
'regretted' is the definition.
('rue' can be a synonym of 'regret')
'impolite we hear' is the wordplay.
'impolite' becomes 'rude' (similar in meaning).
'we hear' indicates a 'sounds like' (homophone) clue.
'rude' sounds like 'RUED'.
'being' is the link.
(Other definitions for rued that I've seen before include "Was contrite about" , "Bitterly regretted" , "'Regretted, repented (4)'" , "was remorseful" , "Was sorry" .)