You didn't say what the adjustments were for, but assuming the employer did pay for some type of moving-related expense in 2023, they are REQUIRED to add it to your W-2. If you are unsure what they paid, then ask your employer for details.
You mentioned expenses, but for most situations moving expenses are no longer deductible on the Federal tax return. So any employer paid moving expenses would be fully taxable regardless if you had expenses or not.
You just report your W-2. There is no need to mention moving, either for 2022 or 2023 (unless your State allows a deduction for moving expenses).