Recently the thought of autogyro flight as "powered autorotation" came to mind. The rotor acts like a fixed wing, providing lift, while a separate source of thrust provides forward motion.
A helicopter tilts its rotor forward a bit using for both lift and thrust.
But once it moves forward, it would seem the change in relative wind through the rotor may improve its lifting efficiency. But, as with fixed wing aircraft, forward motion increases form drag.
What does a helicopter pilot do with the controls when the helicopter accelerates to cruise? How does the rate of fuel consumption change?