Get rid of Republican, keeping quiet about it (5)
I believe the answer is:
erase
'get rid of' is the definition.
(to erase something is to get rid of or remove it)
'republican keeping quiet about it' is the wordplay.
'republican' becomes 'r' (abbreviation).
'keeping quiet' becomes 'ease' (I am not sure about this - if you are sure you should believe this answer much more).
'about it' indicates putting letters inside.
'r' going inside 'ease' is 'ERASE'.
(Other definitions for erase that I've seen before include "Put out of mind" , "Efface" , "Clear" , "Blot out" , "Remove permanently" .)