Religious person, a primate (8)
I believe the answer is:
capuchin
This clue is a double definition.
'religious person' is the first definition.
(I've seen this before)
'a primate' is the second definition.
(capuchin is a kind of primate)
(Other definitions for capuchin that I've seen before include "Friar; cloak and hood" , "Religious person" , "Religious order; woman's pointed hood and cloak resembling the habit" , "Monkey with a head like a monk's cowl" , "China cup for kind of friar" .)