It is possible you owe city/local taxes since you reported the wrong amount. However, often the local tax withheld will equal the actual tax due as the calculation of that is usually simple. I suggest you prepare an amended return in TurboTax so see if you end up owing anything. If not, you don't need to file the amended return. Most likely it won't affect your federal return unless you itemize deductions and deduct your local taxes, but even then the impact would probably be minimal.
You should wait until the return you filed is processed as the state government may adjust your return if necessary as they will have a record of what your local taxes and withheld amount is from your W-2.
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