Primate is upset with bishop, say, touring Italy (6)
I believe the answer is:
simian
'primate' is the definition.
(simian is a kind of primate)
'is upset with bishop say touring italy' is the wordplay.
'upset' shows that the letters should be reversed in order.
'with' means one lot of letters go next to another.
'bishop say' becomes 'man' (bishop is a kind of man).
'touring' indicates putting letters inside (touring can mean going round).
'italy' becomes 'i'.
'is' in reverse letter order is 'si'.
'man' enclosing 'i' is 'mian'.
'si'+'mian'='SIMIAN'
(Other definitions for simian that I've seen before include "Monkey-like" , "Relating to monkeys" , "Ape; apelike" , "Resembling an ape or monkey" , "Ape-like - in Siam (anag)" .)