Regrets most sincere non drinker being thrown out (4)
I believe the answer is:
rues
'regrets' is the definition.
('rue' can be a synonym of 'regret')
'most sincere non drinker being thrown out' is the wordplay.
'most sincere' becomes 'truest' ('true' is associated in meaning with 'sincere').
'non drinker' becomes 'tt' (abbreviation for teetotaller).
'being thrown out' is a deletion indicator.
'truest' with 'tt' taken away is 'RUES'.
(Other definitions for rues that I've seen before include "Regrets, to be sure" , "Bemoans" , "Is sorry for - plants" .)