Insolence regretted, we hear, to Head (8)
I believe the answer is:
rudeness
'insolence' is the definition.
(thesaurus)
'regretted we hear to head' is the wordplay.
'regretted' becomes 'rued' ('rue' can be a synonym of 'regret').
'we hear' shows a homophone (sound like).
'to head' becomes 'ness' (ness is a headland).
'rued' sounds like 'rude'.
'rude'+'ness'='RUDENESS'
(Other definitions for rudeness that I've seen before include "Vulgarity" , "Offensive bad manners" , "Discourtesy" , "Lip" , "loutish behaviour" .)