One bighead lives with a bird (4)
I believe the answer is:
ibis
'a bird' is the definition.
(ibis is a kind of bird)
'one bighead lives' is the wordplay.
'one' becomes 'i' (Roman numeral).
'bighead' becomes 'b' (the head letter of 'big').
'lives' becomes 'is' ('be' can be a synonym of 'live').
'i'+'b'+'is'='IBIS'
'with' acts as a link.
(Other definitions for ibis that I've seen before include "Bird sacred in Ancient Egypt" , "Wading bird of tropical waters" , "Wading bird with a down-curved bill" , "Eastern" , "may be sacred?" .)