Here are some things you may want to try:
1. Clear the Print Spooler.
Sometimes a "ghost" document gets stuck in the brain of the computer's print service, blocking everything behind it. Press Windows Key + R, type services.msc, and hit Enter. Find Print Spooler in the list. Right-click it and select Stop. Open File Explorer and go to: C:\Windows\System32\spool\PRINTERS Delete everything in that folder (don't worry, these are just temporary stuck files). Go back to the Services window, right-click Print Spooler, and select Start.
2. Check the Port Settings
Because this is an older model, Windows sometimes defaults to the wrong "virtual" port during setup. Go to Settings > Devices > Printers & Scanners. Click your HP P2035 and select Manage, then Printer Properties. Click the Ports tab. Ensure a USB or DOT4 port is checked. If it’s set to "LPT1" or a "WSD" port, it likely won't communicate with a physical USB cable.
3. The "Class Driver" vs. "Full Driver"
Since you did a driver update, you might have the "HP Smart" or "Universal" driver, which can be finicky. Try the HP Universal Print Driver (PCL6). It is often more stable than the model-specific installer. Alternatively, try the Windows Update driver: Unplug the USB, go to Printers & Scanners, "Remove Device," then plug the USB back in and let Windows find its own driver without the HP installer software.
4. Direct Connection Check. Avoid Hubs. If the printer is plugged into a USB hub or a monitor's USB port, plug it directly into the back of the PC. These older LaserJets are sensitive to voltage drops in hubs. Try a USB 2.0 Port: If your computer has both blue (USB 3.0) and black (USB 2.0) ports, try the black one. Older hardware sometimes struggles with the high-speed 3.0 handshake.
5. Quick Hardware Test
To prove the printer itself is 100% fine and it’s definitely a Windows issue. With the printer on and ready (Green light), press and hold the Green/Go button for about 5 seconds until the ready light blinks, then release. If a Self-Test/Configuration page prints, the printer hardware is perfect. If it doesn't, we might be looking at a physical paper feed or internal sensor issue.