To comply with legislation, tax calculates using a week 1 / month 1 free pay allowance. This ignores previous pay and tax and calculates on a non-cumulative basis. Non-cumulative means that the employee's year to date figures aren't included in the tax calculation.
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Instead, at week 53, tax is calculated as if it's the first pay run of the tax year. The tax code displayed in Pay & Bill will not change and you should not amend it. Access Pay & Bill follows HMRC rules and calculates tax correctly at week 53.

A P800 is a notice of under or overpaid tax in the year. The reason week 53 may generate a P800 notice is due to the additional week of tax allowance that the employee receives in this extra week.
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Every year an employee receives a tax free allowance based on their tax code. This is divided over 52 weeks, so at the end of the tax year the employee has received their full allowance. When you process a week 53, tax calculates on a week 1 basis, meaning they have 53 weeks of tax allowance.
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HMRC recalculate 53 weeks of earnings against 52 weeks of the annual allowance. They communicate the additional week of tax due through the P800 and claim it back through a tax code adjustment. Alternatively, the employee can settle this balance directly with HMRC.

Click on the below link for the relevant tax year for further information from HMRC. Section 1.13 refers to week 53 payments.
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​2023 to 2024: Employer further guide to PAYE and National Insurance contributions
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​2024 to 2025: Employer further guide to PAYE and National Insurance contributions