Make an effort to accommodate old right-winger (4)
I believe the answer is:
tory
'right-winger' is the definition.
(I know that Tory is a type of right-winger)
'make an effort to accommodate old' is the wordplay.
'make an effort' becomes 'try' (to make an effort is to sincerely try).
'to accommodate' means one lot of letters goes inside another.
'old' becomes 'o' (abbreviation).
'try' enclosing 'o' is 'TORY'.
(Other definitions for tory that I've seen before include "blue" , "type on right? Yes" , "English Conservative politician or Donegal island" , "Traditionalist" , "Politician; tyro (anag.)" .)