Request has delicacy in eastern US city (8)
I believe the answer is:
entreaty
'request' is the definition.
(I know that entreaty is a type of request)
'delicacy in eastern us city' is the wordplay.
'delicacy' becomes 'treat' (synonyms).
'in' means one lot of letters goes inside another.
'eastern' becomes 'e'.
'us city' becomes 'NY' (abbreviation for New York).
'e'+'ny'='eny'
'treat' placed within 'eny' is 'ENTREATY'.
'has' is the link.
(Other definitions for entreaty that I've seen before include "Supplication" , "petition" , "Teeny rat (anag.)" , "Appeal" , "An earnest or emotional request" .)