Why the printer is printing while connecting to the system

I had this issue with my M401n and dne before. These are prone to it due to a problem with the network stack (specifically, WoL; Wake on LAN) feature due to network stack bugs. The issue is HP never tried to fix it because it became an issue once they were fully invested in the HP DS models :-(. There's no true fix for it. That said, it largely affects modern devices, especially newer Apple devices that run iOS 16 or newer, given the age of the AirPrint protocol they use.

That said, as a mitigation, you can adjust the sleep times to keep the printer awake as late as possible. But the big issue is it's not a permanent fix. Unless you manually wake it up, it will still occur once the printer sleeps.

Due to this issue, I had to replace mine with an MS621 because of this (and other issues like the RAM causing processing delays on complex jobs or automatic simplicity done at the RIP). I tried to avoid it for as long as I could but had to give up and replace the hardware. But if it helps, I bought a used machine with a low PC (~13.8k). I now have it up to 14k and change, and it's been a workhorse.
The BIG downside is when I got the MS621 is that the PC SOMEWHAT MATTERS because the fuser is a service item (200k page lifespan). As such, you can't get off quickly with one with 100k+ pages. It's not an issue if you have a maintenance kit on hand and get the unit at a price that matches the need to service it so soon (and can do the service), but don't pick one with a high PC (100k pages or bordering on 100k pages) if it has the 200k maint kit.

I did this on an M426, but this should show you how to change it: