The public-use instructional flight impacted terrain during a hard landing while practicing autorotations. The private pilot stated the certified flight instructor (CFI) pulled the throttle back and she entered an autorotation. The CFI rolled the throttle back to the full on position. The helicopter was leveled and collective was pulled. The helicopter landed hard. The CFI stated that he noticed an excessive sink rate and he applied the throttle with no response. He applied cyclic to help the private pilot level the helicopter and collective to cushion the landing.