Begrudge gift, heading off (6)
I believe the answer is:
Here is my best explanation:
'begrudge' is the definition.
('resent' can be a synonym of 'begrudge')
'gift heading off' is the subsidiary indication.
'gift' becomes 'present' (present is a kind of gift).
'heading off' means to remove the first letter.
'present' with its initial letter taken off is 'resent'.
(Other definitions for resent that I've seen before include "Feel bitterness at (something)", "Hold bitter memory, begrudge", "Feel displeased or indignant about", "Be offended by", "Feel affronted by", "Take exception to", "Feel indignation at", "Begrudge", "Nurse bitter or angry feeling" and "Dispatched again - take amiss".)