I guess some cities actually have the word "city" in their names, e.g. New York City, some others not, e.g. Boston, Beijing.
In some informal situations, people might use "New York", "Boston City" or "Beijing City".
In formal situations, "New York City" would have the word "city"; "Boston", "Beijing" or some other cities like them would not have the word "city".
Is my understanding right?
However, I found some news use "New York City" while some use "New York". Could you give us some examples of the contexts?