Yes, you will need to file Form 8606 on your 2020 tax return to report recharacterization and make the contribution nondeductible. Then you will have a basis on line 14 of your 2020 Form 8606 to carry to 2021.
Yes, you will have to amend your 2020 tax return if you do not already have Form 8606 on your 2020 tax return. Otherwise, the IRS will ask questions if you report the $4,000 from 2020 on your 2021 tax return. And you want to enter the 2020 basis on your 2021 tax return to make your conversion nontaxable (only earnings would still be taxable).
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