In American English, "district" is rarely used except to refer to "school districts"--the maps that determine which schools children attend. In the context of this sentence, we'd be more likely to say "in my area" or "in my neighborhood". Would either of those be acceptable here?
A location within a larger area usually gets the preposition "in".
Glad you added movie theatre as we definitely don't say "cinema" for movie theater in American English, though I think they (maybe?) do in British English. The "district" thing is also weird, but I trust that it's a normal way to refer to neighborhoods in Polish speaking cities.
"Nie ma kina w mojej dzielnicy" could work if you were correcting someone's claim (Let's go to the nearest cinema! There must be a cinema around here, right? - No, there is no cinema in my district.), but generally your version is better in my opinion. I'll remove this sentence and create the one that you mentioned instead.