Yes it is allowed , it can be used to add more specific attention to one specific area of New York City that has been affected.
Say you were writing about COVID-19 and theatres in New York.
That sentence would be perfect to show specifically how COVID-19 has impacted theatres.
ell.stackexchange.com/users/109564/Daniil
The sentence is both grammatical and acceptable.
I agree that it may seem redundant without context.
If New York City was plagued, so (presumably) were New York City's theaters, pizza parlors, offices, recreation centres, brothels, etc.
However, if the article is specifically about theaters in NYC, it adds focus.
It specifies that we are not analyzing the effect of COVID-19 on all of New York City.
It is the theaters that are relevant to the article.
Hope this helps!
ell.stackexchange.com/users/113846/Micah Windsor
If you just say, "COVID-19 plagued New York City," then maybe all the theaters were okay.
(It's extremely unlikely, but possible.)
Maybe they were okay, and maybe they were plagued.
That sentence doesn't tell you.
It just tells you that the city was plagued, but it doesn't tell you which part or parts of the city.
However if you say, "COVID-19 plagued New York City and its many theaters," then theaters definitely were plagued.
However, let's look at another sentence:
The theaters in New York City were plagued, and in New York City, the theaters were plagued.
This sentence is bad style.
It is very redundant.
It sounds kind of stupid.
However English grammar does allow this.
It is not a grammatical problem, but a problem with style.
ell.stackexchange.com/users/4427/Panzercrisis
Geographically, the theaters might be considered part of the city, but in pure logic they are not (and you're talking about logic).
Compare with "Gun crime plagues New York City".
You're making an inference that a disease that plagues the city also plagues its theaters, and you're treating the statement as redundant because you think the reader is capable of making the same inference.
You might be right; but it's an inference that requires knowledge of how diseases work, rather than relying on pure logic.
ell.stackexchange.com/users/9574/Michael Kay
There are a few things to consider here.
Firstly, grammar doesn't generally concern itself with the meaning of a sentence.
There is a famous example by Noam Chomsky of a sentence which is grammatically correct , but nonetheless nonsensical in multiple ways: colorless green ideas sleep furiously .
So the sentence you give is grammatically correct because it is of the form "[noun phrase] plagued [noun phrase] and [noun phrase]", which is allowed by the grammar of English.
Secondly, being redundant doesn't make something nonsensical .
Nonsense would imply that the two parts of the sentence contradict each other, not just that one of them was unnecessary.
So the sentence is not nonsensical either.
Thirdly, if a sentence is technically redundant , that doesn't necessarily mean it is bad style .
Repetition and redundancy are often used as literary or rhetorical devices, e.g. to emphasise a point.
In this example, it is used to establish context, first to the city, and then to "zoom in" to the city's theaters.
Finally, it's worth knowing that the specific structure used in this example, "X and its Y", is very common, and "[city] and its many [item]s" is almost a cliché in its own right.
ell.stackexchange.com/users/9488/IMSoP
This is not redundant.
You might think it is obvious since New York City has many theaters that they were also plagued by COVID-19, but this assumes something about the nature of theaters and of COVID-19.
To see that it is not redundant, compare the following ostensibly similar (but false) statement.
COVID-19
plagued New York City and its many lampposts.
Of course, as others have said, even if it were logically redundant it would not be ungrammatical and might be a useful way of adding emphasis.
ell.stackexchange.com/users/37000/Especially Lime
As a point of contrast to other answers, I might point out that while English is pretty flexible in this case, the OP has a point that a strict reading may not understand elaboration or redundancy, but can imply a relationship between the theaters and, in this case, being "plagued".
So while the other answers are reasonable that a native speaker might comfortably read the sentence as meaning "including" : COVID-19
plagued New York City and its many theaters. ==
>
COVID-19
plagued New York City, including its many theaters. .
. . I would agree that if we had such a sentence about an unfamiliar subject, a native speaker might well take the sentence to imply "because of" : Bedbugs infested Gotham City and its many cramped high-rises. ==
> Bedbugs infested Gotham City (at least in part) because of its many cramped high-rises.
ell.stackexchange.com/users/8696/Mike M
I don't think it breaks any grammar rules, not even rules of logic (where "A and A" is a correct expression).
However, most people would hopefully agree that there are better ways to imply that theatres had a significant role to play in this case.
I would suggest COVID-19
plagued New York City with its many theaters.
ell.stackexchange.com/users/114397/BotReply
I think it makes sense if you add an implied word, "hence" COVID-19 plagued New York City and [hence, ] its many theaters.
If you just said "COVID-19 plagued New York City's theaters", the audience would be wondering why just the theaters.
This is a convenient way to get to the point you're trying to make ("theaters were affected") by starting from a fact ("New York City was affected"), with the link between them being so obvious (New York's theaters are in New York) that it's not worth spending any words on.
ell.stackexchange.com/users/5162/Steve Bennett
For example:   COVID-19 plagued New York City and its many theaters.   I am wondering if "and its many theaters" makes sense, because it's included in "New York City". Is this allowed in English? Because to me it's already implied and thus it's nonsensical.
Is redundant logic ungrammatical?
ell.stackexchange.com
<grammar>
ell.stackexchange.com/questions/246201
ell.stackexchange.com/users/114055/anchorage
Yes it is allowed , it can be used to add more specific attention to one specific area of New York City that has been affected.  Repetition and redundancy are often used as literary or rhetorical devices, e.g. to emphasise a point. The sentence maybe bad style but it is both grammatical and logical. 
The sentence maybe bad style but it is both grammatical and logical. 