Apple used to send out actual Apple Pencil firware updates that went directly to the Pencil with the 1st gen Apple Pencil models, when connected to the Lightning port, but stopped doing these for some years now and there is no way to ever do this with a 2nd gen Apple Pencil other than to issue new 2nd gen Apple Pencil that have had their firmware chip updated at the factory making level.
I use my Apple Pencil more for drawing than for note taking, but still take enough notes with it.
I am not having this issue, as I have put a silicone tip protector over the tip of my Apple Pencil to both help minimise the noise amd shock of the Pencil tip coming in contact with the iPad's screen on/off the screen more regularly when just writing.
I also use a padded writing/drawing glove which, in my own experiences with the Apple Pencil, have pretty much eliminated any glitching due to palm rejection issues.
Also, unlike a student, I do not use my Apple Pencil every single day for many hours at a time, either.
There is really no way to know if this is just a hardware/software issue or combination of both.
Same goes for intermittent, unresponsive touch issues still plaguing some years amd models of iPad Pro/iPad Air models .
No one can pin down whether these issues are hardware, software or a some combination of both.
And it really appears Apple engineering cannot pin these issues down, either!
You could try a drawing/writing glove to see if this helps or not.
This is the drawing /writing glove I have been using for a few years, now, with my own iPad Pro.
https://www.amazon.com/Timebetter-Rejection-Drawing-Sketching-Suitable/dp/B08DKGHYMS/ref=sr_1_21?dchild=1&keywords=drawing+gloves&qid=1611533128&sr=8-21
It's a temporary/kludgey solution, but it may work out better for you.
Another idea, if your Apple Pencil is glitching this badly is maybe to use this drawing/writing glove with a more commonplace lesser battery powered fine tip capacitive stylus that has no pressure or palm rejection ( the glove would act as the palm rejection) and see if writing notes goes better on your iPad.
Many makers of these on Amazon amd the sell for half to less than half price of an Apple Pencil.
Just my own thoughts on alternatives to help get past these Apple Pencil issues.
