Raised stink with US Republicans taking North back (4)
I believe the answer is:
pong
'raised stink' is the definition.
(pong can informally mean a smell)
'us republicans taking north back' is the wordplay.
'us republicans' becomes 'GOP' (Grand Old Party, nickname for US Republican Party).
'taking' is an insertion indicator.
'north' becomes 'N' (abbreviation).
'back' shows that the letters should be reversed in order.
'gop' placed around 'n' is 'gnop'.
'gnop' back-to-front is 'PONG'.
'with' is the link.
(Other definitions for pong that I've seen before include "hum" , "It may make one wrinkle nose" , "In slang, a real stink" , "Horrid smell (informal)" , "Niff" .)