Irish county has right person heading council (5)
I believe the answer is:
mayor
'person heading council' is the definition.
(mayor can mean the head of a council)
'irish county has right' is the wordplay.
'irish county' becomes 'mayo' (county in Ireland).
'has' means one lot of letters go next to another.
'right' becomes 'r' (common abbreviation).
'mayo'+'r'='MAYOR'
(Other definitions for mayor that I've seen before include "Civic head of town or borough" , "Elected city head" , "City government head" , "Elected town/city head" , "Head of a city council" .)