Government of right, for example, at times ignoring outsiders (6)
I believe the answer is:
regime
'government' is the definition.
(I know that government can be written as regime)
'right for example at times ignoring outsiders' is the wordplay.
'right' becomes 'r' (common abbreviation).
'for example' becomes 'eg' (short for 'exempli gratia', 'for example' in Latin).
'at' means one lot of letters go next to another.
'ignoring outsiders' means to look at the middle letters.
The middle of 'times' is 'ime'.
'r'+'eg'+'ime'='REGIME'
'of' is the link.
(Other definitions for regime that I've seen before include "co-ordinated programme" , "Prescribed course" , "Dominion of emigre" , "Form of rule" , "Command" .)