Hi,
The HP ENVY Photo 7820 All-In-One Printer is listed in About AirPrint - Apple Support. This means that the model in question should be compatible with macOS without the need for an additional driver, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router).
Check that your printer is properly connected to the Wi-Fi router.
The connection does not have to be wireless. The HP ENVY Photo 7820 has Ethernet as well. AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections, so you could choose to wire the printer to one of the LAN ports of the Wi-Fi router instead. Ethernet may be more stable.
If there is a problem with AirPrint device discovery, try a full power-cycle sequence (including the router): Power off the Wi-Fi router, the HP printer, the Mac, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the HP printer. Wait a minute or two. Power on the Mac. Finally, power on all other devices/computers.
Reset the printing system on your Mac to solve a problem – Apple Support (UK)
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)
If necessary, an HP Smart app for Desktop (or iOS/iPadOS) may also be useful for setup purposes.
https://support.hp.com/us-en/product/details/hp-envy-photo-7800-all-in-one-printer-series/model/16483130
https://apps.apple.com/us/app/hp-smart-for-desktop/id1474276998
https://apps.apple.com/us/app/hp-smart/id469284907