Expression of regret from guy losing heart after a horseback game (7)
I believe the answer is:
apology
'expression of regret' is the definition.
(synonyms)
'guy losing heart after a horseback game' is the wordplay.
'losing heart' suggests removing the centre.
'after' means one lot of letters go next to another.
'horseback game' becomes 'polo' (I've seen this in another clue).
'guy' with its middle taken out is 'gy'.
'a'+'polo'='apolo'
'gy' put after 'apolo' is 'APOLOGY'.
'from' acts as a link.
(Other definitions for apology that I've seen before include "Excuse" , "Regrets < poor" , "Poor substitute" , "defence" , "A poor example" .)