Republican following one providing support for The House (6)
I believe the answer is:
rafter
'one providing support for the house' is the definition.
'republican following' is the wordplay.
'republican' becomes 'r' (abbreviation).
'following' becomes 'after' (I've seen this before).
'r'+'after'='RAFTER'
(Other definitions for rafter that I've seen before include "traveller on log platform" , "Sloping beam that supports the roof" , "person on floating logs" , "Strong wooden beam for roof" , "Roofing beam" .)