Brad & Greg roughly oust Republicans of troublesome character (3,3)
I believe the answer is:
bad egg
'brad greg roughly oust republicans' is the definition.
I know nothing about this answer so I can't judge whether it can be defined by this definition.
'troublesome character' is the wordplay.
'troublesome' becomes 'bad' (I've seen this in another clue).
'character' becomes 'egg' (I am not sure about this - if you are sure you should believe this answer much more).
'bad'+'egg'='BAD EGG'
'of' acts as a link.
Can you help me to learn more?
(Other definitions for bad egg that I've seen before include "wicked person?" , "Dishonest person like addled ovoid" , "Nasty person like addled ovoid" , "Untrustworthy type (smelling rotten?)" , "Nasty fellow like addled ovum" .)