Vote for Republican on small matter (8)
I believe the answer is:
electron
'small matter' is the definition.
(I've seen this before)
'vote for republican on' is the wordplay.
'vote for' becomes 'elect' (vote for in an election).
'republican' becomes 'R' (abbreviation in US politics).
'elect'+'r'+'on'='ELECTRON'
(Other definitions for electron that I've seen before include "Negatively charged particle" , "Tiny bit of power" , "Negatively-charged particle" , "Charged subatomic particle" , "In small part" .)