Woman chasing my large dog back in Irish town (9)
I believe the answer is:
ballymena
'irish town' is the definition.
(town in Northern Ireland)
'woman chasing my large dog back' is the wordplay.
'woman' becomes 'ena' (feminine given name).
'chasing' says to put letters next to each other.
'large' becomes 'l'.
'dog' becomes 'lab' (short for labrador).
'back' is a reversal indicator.
'my'+'l'+'lab'='myllab'
'myllab' reversed gives 'ballym'.
'ena' put after 'ballym' is 'BALLYMENA'.
'in' is the link.
(Another definition for ballymena that I've seen is " district in Ireland".)