Paying tax on savings interest under £10,000

Hello, in FY 23-24 I earned around £43,000 from full-time employment (with tax charged at source) and around £9,000 from savings interest. This means that I don't need to register for self-assessment, but I know that I do need to pay tax on my savings - how will this happen? Am I correct in understanding that HMRC will also my tax code and I therefore don't need to do anything?

Hi
If you earn interest over your allowance but under £10,000 and are employed or receive a pension, HMRC will change your tax code so you pay the automatically. The code for the current year will be based on how much interest you had in the previous year. 
Further information can be found here: Tax on savings interest

If HMRC has not adjusted the tax code reflecting 2022/2023. and 2023/2024 does this just mean they are busy but will eventually? For each of these years interest received was under £10,000. Thanks

Hi,
As we would need to access your record to check the tax codes you will need to contact our Income Tax team for advice.
Income Tax: general enquiries
Thank you.