it's possible the IRS will catch the error and adjust your return automatically. if not do not file an amended return until the 2021 return is fully processed. then you'll have to amend 2021. you can't report a 2021 1099G on your 2022 return. if the IRS doesn't catch the omission immediately, it's likely it will be caught later on maybe even 2 years later. then you'll have to pay the additional taxes for 2021 + penalties and interest and then amend 2022 to remove the improperly reported 1099G
