Right -- the man's American, and a primate (6)
I believe the answer is:
rhesus
'a primate' is the definition.
(rhesus is a kind of primate)
'right the man's american' is the wordplay.
'right' becomes 'r' (common abbreviation).
'the man's' becomes 'he's' (he refers to a man).
'american' becomes 'us' (abbreviation for United States).
'r'+'hes'+'us'='RHESUS'
'and' acts as a link.
(Other definitions for rhesus that I've seen before include "Kind of money or blood type" , "Small Asian macaque" , "Small Indian monkey" , "Often a positive factor in man" , "King, in Greek mythology" .)