A game, extremely grubby, leads to expression of regret (7)
I believe the answer is:
apology
'expression of regret' is the definition.
('apology' can be a synonym of 'expression of regret')
'a game extremely grubby' is the wordplay.
'game' becomes 'polo' (polo is a kind of game**).
'extremely' suggests removing the centre.
'grubby' with its centre removed is 'gy'.
'a'+'polo'+'gy'='APOLOGY'
'leads to' is the link.
(Other definitions for apology that I've seen before include "Acknowledgment of an offence" , "A poor example" , "Regretful acknowledgement of a fault" , "Saying sorry; poor specimen" , "Expression of regret for offence given" .)