Carrier holding rook or crow (4)
I believe the answer is:
brag
'crow' is the definition.
('brag' can be a synonym of 'crow')
'carrier holding rook' is the wordplay.
'carrier' becomes 'bag' (I've seen this before**).
'holding' is an insertion indicator.
'rook' becomes 'r' (abbreviation in chess).
'bag' enclosing 'r' is 'BRAG'.
'or' is the link.
(Other definitions for brag that I've seen before include "'Show off, boast (4)'" , "boast; form of poker" , "Boast about garb" , "game, making > boast" , "Blow one's own trumpet" .)