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