In your original question you said that New York changed your tax return to include the unemployment that you received. I assume that means that you must have omitted the unemployment when you filed your tax return, which means that you didn't enter your 1099-G at all. If you didn't enter your 1099-G, the state tax withheld was not entered in your tax return. When New York added the unemployment income to your tax return, did they also add the tax withheld in box 11 of your 1099-G? If they did not add the withheld tax, that's why you owe so much now. They are not giving you credit for the tax that was withheld from your unemployment. You have to get them to add the withheld tax. Look in the notice that you got from the state for instructions for what to do if you don't think that their changes are correct.
If you didn't enter your 1099-G when you filed your 2020 tax return, then the unemployment income was also not included in your federal tax return. If the gross unemployment compensation, box 1 of the 1099-G, is more than $10,200, the IRS is going to be asking you for more money too. Even if box 1 is not more than $10,200, if you didn't enter the 1099-G, any federal tax withheld in box 4 of the 1099-G was not included in your federal tax return. So you should file an amended federal tax return to add the 1099-G if box 1 is more than $10,000 or there is any amount at all in box 4. It's better to do that now than to wait until you get a notice from the IRS. If the federal withholding was omitted, you might actually be entitled to an additional federal refund.
If you did enter your 1099-G when you prepared your tax return, why did New York have to add the unemployment?
