After heading off, hitch horse (3)
I believe the answer is:
nag
'horse' is the definition.
(term for an old horse)
'after heading off hitch' is the wordplay.
'after heading off' means to remove the first letter (the letter which heads the word is removed).
'hitch' becomes 'snag' (snagging is a kind of hitching).
'snag' with its first letter taken away is 'NAG'.
(Other definitions for nag that I've seen before include "Bother persistently" , "Badger with constant complaints" , "Jade" , "Persistently worry; horse" , "Ailing horse" .)