city vs administrative district

city and administrative district both are nouns.

Word	Noun	Adjective	Verb	Adverb
city	Yes	No	No	No
administrative district	Yes	No	No	No

As nouns, administrative district is a hypernym of city; that is, administrative district is a word with a broader meaning than city:

Other hypernyms of city include administrative division, territorial division.

city (noun)	administrative district (noun)
people living in a large densely populated municipality	a district defined for administrative purposes
a large and densely populated urban area; may include several independent administrative districts
an incorporated administrative district established by state charter