Bill claims Republican is irritating (6)
I believe the answer is:
grates
'is irritating' is the definition.
'bill claims republican' is the wordplay.
'bill' becomes 'gates' (Bill Gates).
'claims' indicates putting letters inside.
'republican' becomes 'r' (abbreviation).
'gates' going around 'r' is 'GRATES'.
(Other definitions for grates that I've seen before include "Makes rasping sound, hard on ear" , "Rasps against, food or nerves" , "Makes harsh noise by rubbing - on nerves?" , "Makes harsh sound by rubbing" , "Rubs harshly against, perhaps on nerves" .)