Does the HP EliteDesk 800 G9 Mini support 12th Gen Intel processors?

No, the HP EliteDesk 800 G9 Mini only supports 13th Gen Intel Core processors (Raptor Lake) on LGA 1700 — i3, i5, i7, and i9 at 35W or 65W TDP. Unlike most LGA 1700 systems, HP locked it to 13th Gen only, so 12th Gen or earlier CPUs won’t boot.