Church primate starts reading out notices from The Guardian (8)
I believe the answer is:
chaperon
'the guardian' is the definition.
(chaperon is a kind of guardian)
'church primate starts reading out notices' is the wordplay.
'church' becomes 'ch' (common abbreviation**).
'primate' becomes 'ape' (I've seen this before).
'starts' suggests taking the first letters.
The first letters of 'reading out notices' is 'ron'.
'ch'+'ape'+'ron'='CHAPERON'
'from' is the link.
(Other definitions for chaperon that I've seen before include "Attendant" , "One accompanies to supervise" , "Duenna; look after" , "Person accompanying girl or younger group" , "Preach on about older woman in charge of girl" .)