Thank you for the information.
The HP OfficeJet Pro 9025e All-In-One Printer is listed in About AirPrint - Apple Support as a printer supporting the driverless AirPrint technology. An AirPrint-enabled printer is compatible with macOS (13 Ventura), and can be used without additional drivers, provided that the printer and the Mac are connected to the same network (Wi-Fi router). AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections (so, you could connect this printer by Ethernet cable to one of the LAN ports of the router instead of using Wi-Fi, if you so wish).
The aforementioned power-cycle procedure is intended to ensure that all devices are correctly registered with the router (that is why the router is included in the procedure).
AirPrint will work as it is, without extra software. What complicates the matter is that HP seems to require the use of an HP Smart app, and an always active (!) Internet connection, for ink services.
Sorry, I do not have any other ideas regarding the certificate issue.
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)
Reset the printing system on your Mac to solve a problem – Apple Support (UK)
If you have access to an iPhone/iPad, does AirPrint printing work from there (via the Wi-Fi router), without an extra app?
Use AirPrint to print from your iPhone, iPad, or iPod touch - Apple Support