Monarch that's rejected in this republic (4)
I believe the answer is:
eire
'this republic' is the definition.
(Irish name for Ireland)
'monarch that's rejected' is the wordplay.
'monarch' becomes 'er' (abbreviation for Elizabeth Regina).
'that's' becomes 'ie'.
'rejected' says the letters should be written in reverse.
'er'+'ie'='erie'
'erie' written backwards gives 'EIRE'.
'in' is the link.
(Other definitions for eire that I've seen before include "Irish name for Ireland" , "island" , "Gaelic name for Republic of Ireland" , "Southern Ireland" , "Former name of the Irish Republic" .)