Moving expenses are, and always have been, a personal expense. If an employer pays those expenses for you, the amount of the payments, directly to you or to anyone else on your behalf, is taxable income. In tax years prior to 2018, that additional income was offset by your ability to take an adjustment to income for moving expenses as long as certain time and distance tests were met.
Beginning in tax year 2018, the Tax Cuts and Jobs Act of 2017 eliminated the adjustment for moving expenses for all taxpayers except military personnel having a permanent change of duty station.
The $10,000 which you said was sent to the tax authorities would have been additional tax withholding, paid to you as taxable income but sent to the IRS and/or state on your behalf, so that you would not face an enormous tax bill. The "credit" you are seeking will be seen in your tax return when that additional tax withholding is there to offset the loss of the moving expense adjustment.
It sounds like you have a very generous employer.
