Primate accepts vegetables and daughter is mollified (8)
I believe the answer is:
appeased
'is mollified' is the definition.
(I've seen this in another clue)
'primate accepts vegetables and daughter' is the wordplay.
'primate' becomes 'ape' (large primate).
'accepts' means one lot of letters goes inside another.
'vegetables' becomes 'peas' (peas are a type of vegetable).
'and' says to put letters next to each other.
'daughter' becomes 'd' (genealogical abbreviation).
'ape' enclosing 'peas' is 'appease'.
'appease'+'d'='APPEASED'
(Other definitions for appeased that I've seen before include "Placated, pacified" , "Changed a deep sap and mollified" , "Conciliated" , "Assuaged, mollified" , "Calmed" .)