Greek Primate supplying fruit (5)
I believe the answer is:
grape
'fruit' is the definition.
(grape is a kind of fruit)
'greek primate' is the wordplay.
'greek' becomes 'gr'.
'primate' becomes 'ape' (I've seen this before).
'gr'+'ape'='GRAPE'
'supplying' is the link.
(Other definitions for grape that I've seen before include "Vineyard fruit" , "It grows on the vine" , "Small fruit; type of shot" , "Green/black fruit" , "Fruit from bunch" .)