iMac High Sierra will not recognize my HP Office Jet Pro 8610 printer

First try another cable to see if it makes a connection. If it does not work, then please try restarting the computer in Safe Mode and then restart normally and test to see if the printer prints. If it does not then remove the printer in System Preferences and then re-add it.

Hi,

Also, since your printer appears to be a model supporting Apple's AirPrint technology (About AirPrint - Apple Support), it should work with macOS 10.13 High Sierra without any additional drivers, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router).

However, the printer manufacturer may or may not provide a special app for setup and ink management purposes, possibly with higher operating system requirements. Thus, a manual setup may be needed.

Is the phone that you mentioned an iPhone? If so, are you already using AirPrint, or do you print via an iOS app from the printer manufacturer?

AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections, so you could choose to wire the printer by Ethernet cable to one of the LAN ports of the router instead of using Wi-Fi. Ethernet may be more stable and could eliminate some of the issues related to a wireless connection.

A USB connection would require compatible drivers from the printer manufacturer.