Everyone has a tax-free allowance. The year 2025/2026 is £12570.
When you have two or more jobs on a payroll, the tax-free allowance is usually applied to the job where you earn the most amount of money. However, you can ask HMRC to split that allowance against both jobs - I am not sure on the ratio of the split, for example, may be it could be £6285 against each job, no matter what your earn.
If you don't split your allowance, the first £12570 of income on your "main" job won't be taxed. All of your income on the second job will be taxed immediately. However if you know that you'll earn more than £12570 in your main job anyway, it is all as broad as it is long, because you'll pay the same amount of tax whichever way you look at it, it's just a case of when. And every year it will all correct itself, and you'll be paid back anything you are owed.
In the past, I've done several jobs on payrolls all at the same time, and I have never chosen to split my allowance - I just left it against the "main" job, and paid full tax on the others, because as I said above, it all amounts to the same thing, it's just that some people don't like to pay their taxes any sooner than they have to.
It does get more complicated (but not much more) when you are on a payroll and are also working as self-employed & do a tax return, as the self-assessment is usually for money earned many months -even over a year- earlier, so the time-frames for when you earned on a payroll and earned as self-employed are not in harmony. I use an accountant to work all this out for me anyway, but I always opted to pay the full tax on my payroll earnings as I earned them, which was then off-set against my self-employed earnings for the same period, when the self-assessment eventually went in.
You ask if you should inform HMRC, and I would say that in your position I would, just so you know exactly what your options are.