Yes, you can always pay estimated taxes. Since it is the beginning of the year, you should speak with your HR and get them to correct your payroll records for 2021 so that you don't have to deal with paying estimated taxes to NY. They should be able to deduct and pay your taxes directly to the states involved, which if you do that now, you won't have an issue come year end.
For 2020, it is probably too late to affect any penalty that has already been incurred since payments are based on quarters and you have nothing paid in the first 3 quarters. You can still make a payment today or tomorrow that would cover the 4th quarter which may help. Here is the website that you can do that at - Make a NY tax payment
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