Go round denying Republican's remarks on passing (4)
'remarks on passing' is the definition.
(I've seen this before)
'go round denying republican's' is the wordplay.
'go round' becomes 'orbit' (I've seen this before).
'denying' is a deletion indicator.
'republican' becomes 'r' (abbreviation).
'orbit' with 'r' taken away is 'OBIT'.
(Other definitions for obit that I've seen before include "late anniversary", "Short report of perishing", "Death notice in paper", "Remarks on passing", "The last word".)