New York district some hope to cut in half (4)
I believe the answer is:
soho
'new york district' is the definition.
'soho' can be an answer for 'district' (I have seen 'District of London ' mean 'soho' so perhaps 'district' could also mean 'soho'). I am not certain of the 'new york' bit.
'some hope to cut in half' is the wordplay.
'some hope' becomes 'soh' (I can't justify this - if you can you should give a lot more credence to this answer).
'cut in half' suggests halving the letters.
'to' cut in half is 'o'.
'soh'+'o'='SOHO'
Can you help me to learn more?
(Other definitions for soho that I've seen before include "capital district" , "Huntsman's call - area of London" , "Once notorious part of central London" , "London's former Bohemia" , "Exotic London district popular with tourists" .)