Feel nostalgia for half of US state (4)
I believe the answer is:
miss
'feel nostalgia for' is the definition.
(I've seen this in another clue)
'half of us state' is the wordplay.
'half of' indicates taking half.
'us' becomes 'me' ('us' can mean 'mean' in some UK dialects).
'state' becomes 'iss' (I am not sure about this - if you are sure you should believe this answer much more).
'me' halved is 'm'.
'm'+'iss'='MISS'
Can you help me to learn more?
(Other definitions for miss that I've seen before include "Overlook - misfire - young lady" , "One is as good as a mile" , "Girl or unmarried woman" , "Fail to hit (something)" , "Omit" .)