Begrudge gift, heading off (6)
I believe the answer is:
resent
'begrudge' is the definition.
('resent' can be a synonym of 'begrudge')
'gift heading off' is the wordplay.
'gift' becomes 'present' (present is a kind of gift).
'heading off' means to remove the first letter.
'present' with its first letter taken away is 'RESENT'.
(Other definitions for resent that I've seen before include "grudge" , "Feel bitter or indignant about" , "Hold bitter memory, begrudge" , "Feel bad about" , "Feel displeased or indignant about" .)