Primate giving blessing to graduate (6)
I believe the answer is:
baboon
'primate' is the definition.
(baboon is a kind of primate)
'blessing to graduate' is the wordplay.
'blessing' becomes 'boon' (synonyms).
'to' says to put letters next to each other (I've seen this in other clues).
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'boon' after 'ba' is 'BABOON'.
'giving' acts as a link.
(Other definitions for baboon that I've seen before include "clumsy uncouth person" , "Large long-nosed Arabian or African monkey" , "Large primate" , "A large one of 17 across" , "Type of large monkey" .)