Start with the switch - when the ball goes into the HR hole it will close a switch -- clean and properly gap that to 1/8 inch or less. If not that then trace where that wire goes (use a schematic) as it will trigger a relay or something to make the Running Man Unit to activate. Check the wire for breaks or loose solder joints too.
First - do the other targets work correctly and send a man around the bases? And score correctly? Then its the HR switch or something downstream from it