Boast of British newspaper (4)
I believe the answer is:
brag
'boast' is the definition.
(I know that boast can be written as brag)
'british newspaper' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'newspaper' becomes 'rag' (informal term for a newspaper).
'b'+'rag'='BRAG'
'of' acts as a link.
(Other definitions for brag that I've seen before include "'Show off, boast (4)'" , "Boast about garb" , "Card game, like poker" , "Talk boastfully" , "game, making > boast" .)