Primate dismissing bishop perhaps as bore (5)
I believe the answer is:
drill
'bore' is the definition.
(bore can mean to drill into something)
'primate dismissing bishop perhaps' is the wordplay.
'primate' becomes 'mandrill' (mandrill is a type of primate).
'dismissing' indicates named letters should be taken away.
'bishop perhaps' becomes 'man' (bishop is a kind of man).
'mandrill' with 'man' removed is 'DRILL'.
'as' is the link.
(Other definitions for drill that I've seen before include "Put a hole in" , "Mollusc that preys on oysters" , "Dental assistant" , "Make a hole with a pointed tool" , "Rehearsal of the procedure to be followed in an emergency" .)