Boast made by head of broadsheet paper (4)
I believe the answer is:
brag
'boast' is the definition.
(brag is a kind of boast)
'by head of broadsheet paper' is the wordplay.
'by' means one lot of letters go next to another.
'head of' suggests taking the first letters.
'paper' becomes 'rag' (rag is a kind of paper**).
The initial letter of 'broadsheet' is 'b'.
'b' next to 'rag' is 'BRAG'.
'made' is the link.
(Other definitions for brag that I've seen before include "Show off, crow" , "Card game, like poker" , "Boast ? card-game" , "popular with 15 [EGOTIST] , no doubt" , "Shoot a line" .)