Grados will hum with certain 'tables, like the Rega. Usually this comes from the table not being sufficiently grounded. For example, the Rega needs to have its motor and platter bearing both grounded to the arm ground. I suspect the same is true in this case, based on the fact that if you move the arm, the hum changes. That's not a ground loop!
So this might be easy to fix- have a wire installed that goes from the motor chassis to the platter bearing and then to the base of the arm- make sure that every bit of metal on the 'table is grounded.
So this might be easy to fix- have a wire installed that goes from the motor chassis to the platter bearing and then to the base of the arm- make sure that every bit of metal on the 'table is grounded.