If this contribution made in 2024 was properly recorded by the HSA custodian as a contribution for 2023, in the HSA contribution section of 2024 TurboTax answer Yes when asked if your employer told you about other contributions, then enter the $382.08 in the box for contributions made in 2024 for tax year 2023.
Note that you needed to do something similar with your 2023 tax return where you would need to have answered Yes when asked if your employer told you about other contributions and then entered the $382.08 in the box for contributions made in 2024 for tax year 2023, otherwise the $382 would not have appeared on your Form 8889 as required.
You mentioned making a catch-up contribution. If by that you mean that you were age 55 or over in 2023, your contribution limit was $4,850 and you made no excess contribution. If you were under age 55 at the end of 2023, you made no catch-up contribution, just an excess contribution of $178.22. Correcting the excess contribution now requires making a taxable HSA distribution, reportable on your 2025 Forms 8889 and 5329. Avoiding the excess contribution penalty for 2024 would have required that the corrective distribution be made in 2024. The result will a second taxation of the $178.22, plus a 20% additional tax.
