My Apple Pencil 2nd gen stops working after a few minutes of use

Your Apple Pencil is exhibiting symptoms of a failing battery - that will not reliably accept or hold any charge.

You can see the state of the Pencil's battery charge in iPad settings:

Settings > Apple Pencil

You can also install a Battery Widget on any of your Home Screens - or the slide-over Today view.

Add, edit, and remove widgets on iPad - Apple Support

It is essential that if you have an Apple Pencil that you charge it regularly - whether used or not - so as to protect the battery from deep-discharge. Do not allow a pencil to remain in low-charge state for any period of time - as the internal battery will fail, rendering the Pencil useless. Setting aside an unused Pencil, for extended periods, is a recipe for premature death of the Pencil battery.

All rechargeable batteries will eventually fail, even where maintained optimally throughout their serviceable lifespan.

If the Pencil Battery has failed, the only remedy is to replace the Pencil. If the battery has failed and your pencil is within its one-year warranty, you should look to having it replaced by your retailer or at an Apple Store.

You would be best 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. If a fault is identified, you’ll be in the best possible place to explore your available options:

Genius Bar Reservation and Apple Support Options - Apple

Your screenshot shows the battery state while the Pencil is docked with the iPad to charge.

What happens to the indicated state of charge when you remove the Pencil from the inductive charging dock? Does the indicated battery charge quickly/suddenly fall? If yes, this would suggest that the Pencil's battery is failing.