Feel remorse having run over bird (6)
I believe the answer is:
regret
'feel remorse' is the definition.
(I've seen this before)
'run over bird' is the wordplay.
'run' becomes 'r' (cricket abbreviation).
'over' means one lot of letters go next to another.
'bird' becomes 'egret' (egret is a kind of bird).
'r'+'egret'='REGRET'
'having' acts as a link.
(Other definitions for regret that I've seen before include "Bad feeling" , "Rue, repine" , "Be sorry for an action" , "Subsidiary theorem" , "Mourn" .)