Begrudge press agent releasing gasp (6)
I believe the answer is:
resent
'begrudge' is the definition.
('resent' can be a synonym of 'begrudge')
'press agent releasing gasp' is the wordplay.
'releasing' indicates named letters should be taken away.
'press'+'agent'='pressagent'
'pressagent' with 'gasp' removed is 'RESENT'.
(Other definitions for resent that I've seen before include "Hold bitter memory, begrudge" , "Feel aggrieved at" , "Feel affronted by" , "Feel displeased or indignant about" , "Feel bitter or envious" .)