Bishop displacing Nazis in instrument for primate (6)
I believe the answer is:
baboon
'primate' is the definition.
(baboon is a kind of primate)
'bishop displacing nazis in instrument' is the wordplay.
'bishop' becomes 'B' (abbreviation used in chess).
'displacing nazis' becomes 'oo' (I am not sure about this - if you are sure you should believe this answer much more).
'in' means one lot of letters goes inside another.
'instrument' becomes 'ban' (ban is a kind of instrument).
'b'+'oo'='boo'
'boo' put into 'ban' is 'BABOON'.
'for' is the link.
(Other definitions for baboon that I've seen before include "wild swinger?" , "Stupid person" , "A large one of 17 across" , "Type of large monkey" , "Large long-nosed Arabian or African monkey" .)