Your iPad Pencil tip should be changed as soon as visible wear, however slight, becomes evident. Ensure that the Pencil tip is screwed firmly into place; if only slightly loosened, the Pencil may not work properly - if at all. When correctly tightened, there should be tiny gap (of about the thickness of a sheet of paper) between the Pencil tip and the main body of the Pencil.
While some screen protectors are known to interfere with reliable operation of an Apple Pencil, if the Pencil was previously working without issues, presence of the screen protector should not alter performance over time - unless the Pencil itself has an issue.
If changing the Pencil tip does not resolve the problem, it is perhaps possible that the Pencil’s tiny rechargeable battery is reaching the end of its useful life. Do you need to recharge the Pencil more frequently - and/or does the indicted battery status fall rapidly while being used?
You might be well advised to visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to test and assess your iPad and Pencil:
Genius Bar Reservation and Apple Support Options - Apple