Employer made overpayment - how to correct p60

I started a new employment in March 2024. I am entitled to tax free child-care, as my salary is below £100,000. My employer overpaid me in March 2024 (by mistakenly recording me as a full time, rather than part time, employee). The overpayment will be re-paid, likely in April 2024. The overpayment may artificially inflate my salary above £100,000 for 2023/24. HMRC advised that my employer can correct the tax position by submitting a Full Payment Submission/Earlier Year Update. My employer has informed me that they will not correct my P60 because income tax is calculated when a payment is made, rather than when it may or may not have been due. If not corrected, I am concerned that this means that (1) HMRC will consider that I am ineligible for support with childcare costs in 2024/25, and (2) it will appear to HMRC that I made a false income declaration re. my 2023/24 eligibility for support with childcare costs (potentially resulting in clawback). Please can you advise: (i) if my employer is taking the correct steps (including by saying that income tax is calculated when a payment is made rather than when it may or may not have been due), and if not, (ii) what I can do to encourage them to rectify my tax position. Thank you.

Hi Seeking Help,
If your employer has incorrectly overpaid you, and this will have an impact on your tax-free childcare, they should be doing everything they can to help you resolve this issue.
Please ask them to contact us via the Employer's helpline on 0300 200 3200 so we can advise them further on how to rectify this for you.
Employers: general enquiries
Thank you.