precinct vs city district

precinct and city district both are nouns.

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

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

precinct: a district of a city or town marked out for administrative purposes
city district: a district of a town or city

precinct (noun)	city district (noun)
a district of a city or town marked out for administrative purposes	a district of a town or city