Yes, you will have to amend your 2021 return. No, you cannot add it to your 2022 tax return.
You have to report it on your 2021 return no matter if you cash the check or not. These were pre-tax excess contributions and they will have to be added to your wages in 2021. "You must include the excess deferral in your income for the year of the deferral" (Pub 525).
You will have to add the 2022 Form 1099-R with code P to your 2021 tax return. Please see How do I amend my federal tax return for a prior year?
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