The answer to your question is that a signing bonus from a company, for a job that will be based in New York City, is considered New York-source taxable income. Thus, it should be reported on a New York state tax return. Depending on your residency status at the time you actually receive the bonus payment, that New York tax return could be either a resident return, part-year return, or a nonresident return.
If you are still a legal resident ("domiciled" at it is called under the tax laws) in your current state (and not New York) at the time you receive the bonus, you will also have to pay taxes on the bonus in your current state as well. Thus, you would have a situation of both states taxing the same income. Fortunately, that double-taxation is typically significantly reduced, or is eliminated entirely, by the ability to take a state tax credit for taxes paid to another state on double-taxed or "mutually taxed" income. So, it that sense, you're effectively paying taxes on the bonus to just one state . . . even though you could have a filing requirement in two states.
One other factor to consider is that if you agree to accept the job in NYC, but ask your company to delay constructively paying you the bonus check until after you were to actually move to New York state, then you would only have to include the bonus amount on your New York tax return.
Thank you for asking this important question.