Class G just means changing the power (voltage) rails in two or more discrete steps. Lower voltage for lower power means less heat and wasted energy; then, higher voltage(s) allow greater output power when needed. The basic amplifier is usually a class AB amplifier so its low distortion is a function of good design. Class H allows the power rails to vary continuously, increasing as the signal increases and vice-versa, so again saves power. With class G one must ensure there are no artifacts induced when switching the power rails; class H moves the voltage smoothly but is generally more complex and requires greater attention to ensuring the power supply is stable as it varies. Either design can yield excellent performance.
I do not know either intrinsically lends itself to lower distortion in the core amplifier, but either would allow you to run more bias current without as much wasted energy (heat) as a conventional (fixed-rail) design, so it is possible.