Boast made by head of broadsheet paper (4)
I believe the answer is:
Here is my best explanation:
'boast' is the definition.
(brag is a kind of boast)
'head of broadsheet paper' is the subsidiary indication.
'head of' says to take the initial letters.
'paper' becomes 'rag' (rag is a kind of paper**).
The initial letter of 'broadsheet' is 'b'.
'made by' is the link.
(Other definitions for brag that I've seen before include "boast; form of poker", "Card game, like poker", "Talk boastfully", "Crow", "Boast, show off", "Show off, crow", "Idle talk", "Blow one's own trumpet", "Boast, exaggerate" and "popular with 15 [EGOTIST] , no doubt".)