Primate beginning to bestow a blessing (6)
I believe the answer is:
baboon
'primate' is the definition.
(baboon is a kind of primate)
'beginning to bestow a blessing' is the wordplay.
'beginning to' suggests taking the first letters.
'blessing' becomes 'boon' (synonyms).
The first letter of 'bestow' is 'b'.
'b'+'a'+'boon'='BABOON'
(Other definitions for baboon that I've seen before include "Stupid person" , "Type of large monkey" , "Large primate" , "clumsy uncouth person" , "A large one of 17 across" .)