There is not an unemployment compensation exclusion on a New York state income tax return. No amendment is needed.
Go to this New York state website for information - https://www.tax.ny.gov/pit/cares-act-faq.htm#expanded-content-menu10
A number of states changed their tax laws to conform to the federal tax law that excluded the first $10,200 of your 2020 unemployment from being taxed. New York is not one of those states. New York is still taxing ALL of the unemployment you received in 2020. The IRS will refund federal tax to people who filed before the federal tax law changed if you are eligible for the federal "extra" refund.
The IRS is recalculating refunds for people whose AGI is $150K or below and who filed before the tax law that changed the amount of unemployment that is taxable on a federal return became effective.
The IRS began to send out the additional refund checks for tax withheld from unemployment in May. The refunds are being sent out in batches—starting with the simplest returns first. If you are eligible for the extra refund for federal tax that was withheld from your unemployment the IRS will be sending you an additional refund sometime during the next several months. TurboTax cannot track or predict when it will be sent. The IRS has not provided a way for you to track it, so all you can do is wait for the refund to arrive.
Please read this very recent news from the IRS: