Bird having rook trouble (4)
I believe the answer is:
rail
'bird' is the definition.
(a wading bird)
'rook trouble' is the wordplay.
'rook' becomes 'R' (chess abbreviation).
'trouble' becomes 'ail' (synonyms).
'r'+'ail'='RAIL'
'having' acts as a link.
(Other definitions for rail that I've seen before include "The Iron Horse road" , "Means of transport" , "Bar of rolled steel" , "Line" , "Kind of transport" .)