Have you tried to reset your printing system? (System Preferences - Printer&Scanner - Rightclick on the Printer Window / All printer will be removed during this reset, and must be configured again!!)
Can you see the printer in the System Preferences if you select the add a printer menu?
If you see the printer there, have you tried to install it as a (I think it is called in english) legacy printer, so without specific HP-software?
How is the printer connected? If it is via a network, could you reach the printer via its IP-Adress?
Have you installed the EasyStart and EasyAdmin software from this website?
Are you already on macOS 13.1?