Made a judgement not much appreciated about Republican (10)

'made a judgement' is the definition.
(to arbitrate is to judge or decide something)

'not much appreciated about republican' is the wordplay.
'not much' becomes 'a bit' (a bit can mean a small amount).
'appreciated' becomes 'rated' (to rate is to appreciate or commend).
'about' indicates putting letters inside.
'republican' becomes 'r' (abbreviation).
'abit'+'rated'='abitrated'
'abitrated' enclosing 'r' is 'ARBITRATED'.