Employed but no employment history displayed.

Hi, I've been working part-time as an international student on a student visa and I'm not supposed to be taxed. I've been working since the end of September (2023) and I'm still being taxed for every payday. When I contacted HMRC to change my tax code, they said there's no employment history under my Name and NI number. HMRC said that can I contact my employer or HMRC itself could add the employment manually. If I should choose the latter what details who you need to add my employment manually?

Hi Christy,
We would require your employer's PAYE reference number, along with your own works or payroll number, and the date you started working for this employer.
Thank you.