Rushes round to see primate (6)
I believe the answer is:
rhesus
'primate' is the definition.
(rhesus is a kind of primate)
'rushes round' is the wordplay.
'round' indicates an anagram.
'rushes' is an anagram of 'RHESUS'.
'to see' acts as a link.
(Other definitions for rhesus that I've seen before include "Type of monkey and blood category" , "Kind of monkey (or factor)" , "The Rh in "Rh factor"" , "Type of monkey and a blood component" , "Small Indian monkey" .)