Puffed up blood vessel we hear (4)
I believe the answer is:
vain
'puffed up' is the definition.
(I've seen this in another clue)
'blood vessel we hear' is the wordplay.
'blood vessel' becomes 'vein' (vein is a kind of blood vessel).
'we hear' shows a homophone (sound like).
'vein' sounds like 'VAIN'.
(Other definitions for vain that I've seen before include "Swollen-headed" , "Forever looking in the mirror" , "Concieted or pointless" , "is full of himself" , "Conceited, puffed-up" .)