Begrudge suppressing desire briefly on the rise (9)
I believe the answer is:
resurgent
'on the rise' is the definition.
'begrudge suppressing desire briefly' is the wordplay.
'begrudge' becomes 'resent' ('resent' can be a synonym of 'begrudge').
'suppressing' is an insertion indicator.
'desire' becomes 'urge' (I've seen this before).
'briefly' means to remove the last letter.
'urge' with its final letter taken away is 'urg'.
'resent' enclosing 'urg' is 'RESURGENT'.
(Other definitions for resurgent that I've seen before include "Coming back to life" , "reborn" , "Reviving after a lull" , "given new life" , "Reviving in popularity" .)