Post Office New York state’s used for letter delivery system once (4,7)
I believe the answer is:
pony express
'for letter delivery system once' is the definition.
(historical postal system in the US)
'post office new york state's' is the wordplay.
'post office' becomes 'po'.
'new york' becomes 'ny'.
'state' becomes 'express' (synonyms).
'po'+'ny'+'express'='PONY EXPRESS'
'used' acts as a link.
(Other definitions for pony express that I've seen before include "Western film" , "Carrier of mail in the old American West" , "American postal service in the West, 1860-61" , "mounted relay" , "System once used to deliver" .)