My employer, a USG agency, provides a student loan repayment incentive for which they send funds directly to my loan provider. In 2018, they sent more to the loan provider than I owed in loans. The loan provider refunded the overpayment to the USG. Meanwhile, my employer had withheld taxes based on the amount of the original (overpaid) incentive, not the adjusted post-refund amount. They issued W-2 that showed the initial (overpaid) incentive amount as part of my income and the taxes withheld on that basis. After the tax deadline, they issued a corrected W-2 that reflected the correct (post-refund) incentive amount and what the corresponding lower taxes should be. I got an extension and am now preparing to file my return. Do I file using the corrected W-2. And how do I recover the over-withheld taxes?
Your "corrected" W-2 is still wrong. It should show the correct incentive amount (which it does), but the total amount of tax that was actually withheld, even if it was more than it should have been. In other words, the tax withheld on the corrected W-2 should be the same as on the original W-2, because that's the amount that they actually withheld. Reporting the full amount that was actually withheld is how you would recover the over-withholding.
There are two ways to fix this. One is to get your employer to issue another corrected W-2 with the correct income and the correct tax withheld. The other way is for them to just give you a check for the amount of the excess withholding, and not treat the payment as wages, and not include it in your W-2 for either year. It can't be a payroll check.
If they refuse to do either of those, there's a way to indicate on your tax return that your W-2 is incorrect, and use the correct figures. But you first have to try to get the employer to correct it.
In any case, when you file your tax return you have to use the most recent corrected W-2 that you got.