Monarch has regret about Left and Right (5)
I believe the answer is:
ruler
'monarch' is the definition.
(thesaurus)
'regret about left and right' is the wordplay.
'regret' becomes 'rue' ('rue' can be a synonym of 'regret').
'about' is an insertion indicator.
'left' becomes 'l' (common abbreviation).
'and' means one lot of letters go next to another.
'right' becomes 'r' (common abbreviation).
'rue' enclosing 'l' is 'rule'.
'rule'+'r'='RULER'
'has' is the link.
(Other definitions for ruler that I've seen before include "Straight line could come from this" , "Royal" , "Queen, perhaps" , "Governor or drawing stick" , "One exercises dominion" .)