It is likely the IRS will catch the W2 that you left off your return and make the adjustment for you and send you a notice it has been adjusted. You may not need to do anything if this is the case.
If they do not catch this error then you should file an amended return and include this W2.
Before filing an amended return, make sure the IRS has already processed the tax return you need to amend. That way, you can ensure the IRS won't get your original return and amended return mixed up. If you've already received your tax refund, then you know the IRS has already processed your return.
Regarding the adjustment due to the Unemployment Exclusion, there is no need to file an amended return (Form 1040-X) to figure the amount of unemployment compensation to exclude. The IRS will refigure your taxes using the excluded unemployment compensation amount and adjust your account accordingly. The IRS will send any refund amount directly to you.
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How to Correct Federal Tax Returns - TurboTax Tax Tips ...