Regretted having rushed outside? (4)
I believe the answer is:
rued
'regretted' is the definition.
('rue' can be a synonym of 'regret')
'rushed outside?' is the wordplay.
'outside?' means to remove the middle letters (I've seen 'outside' mean this).
'rushed' with its centre removed is 'RUED'.
'having' is the link.
(Other definitions for rued that I've seen before include "Was sorry" , "'Regretted, repented (4)'" , "Bitterly regretted" , "Was contrite about" , "was remorseful" .)