Pacify primate pinching pulses (7)
I believe the answer is:
appease
'pacify' is the definition.
('appease' can be a synonym of 'pacify')
'primate pinching pulses' is the wordplay.
'primate' becomes 'ape' (I've seen this before).
'pinching' is an insertion indicator (taking inside).
'pulses' becomes 'peas' (I've seen this before).
'ape' going around 'peas' is 'APPEASE'.
(Other definitions for appease that I've seen before include "Make peace with" , "Quiet" , "Mollify, placate" , "Mollify or calm by making concessions" , "Placate by making concessions" .)