After doing some reading I think I can explain the gist of it.
Class G and H are modifications to a class AB amplifier's output stages to improve its efficiency and reduce wasted heat/electricity.
The modifications are to the output stage rail voltages, so technically not in the signal path(?)
Class G: Several rail voltages are available, and the lowest one that will do the job at any given moment is selected. For example, during a quiet music passage the rail voltages are +/- 10 V, when the music gets louder the rails switch to +/- 20 V, and at the loudest levels +/- 40 V is selected.
Class H: The rail voltage is a continuously variable analog voltage. Based on the input music signal, just enough voltage is supplied to the output devices for them to reproduce an amplified version of the input.
Class H is more efficient than class G, but also more complex.
https://en.wikipedia.org/wiki/Power_amplifier_classes