If the payment error occurred in 2021, you pay them back what they overpaid you, but they do not make any tax adjustments and they send you a corrected W2 (W2-C) then you will have lower income and more taxes withheld than you needed on the income you actually received.
When you enter your W2-C, your taxes withheld from them will be added to your taxes withheld from your other employer to increase the total amount that you have already paid which will either result in a larger refund or a lower tax due. You will not see the extra amount withheld listed separately anywhere. However, in the end, returning the overpayment, with no tax adjustments will result in a better tax situation for you when you file your return.
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