Arsenal supporter heading off into New York women's house (7)
I believe the answer is:
nunnery
'house' is the definition.
(nunnery is a kind of house)
'arsenal supporter heading off into new york women's' is the wordplay.
I cannot really see how this works, but
'heading' could be 'nne' (I've seen this in another clue) and 'nne' is found in the answer.
'new' could be 'n' (common abbreviation eg NT for New Testament) and 'n' is present in the answer.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for nunnery that I've seen before include "sisters' place" , "Hamlet to Ophelia: 'Get thee to a . . . . . . .'" , "Religious establishment" , "Female religious house" , "Convent" .)