Ngauruhoe and the Mangatepopo Valley, D.Townsend / GNS Science
The high slope and ridge close by to the north is mostly moraine from the last glaciation about 17,000 years ago, with a few thin lavas in it. These were kept out of the valley by the same thick glacier than formed the moraine.
A similar process formed Pukekaikiore but quite a bit earlier, at least 140,000 years ago during the second-to-last glaciation. In both glaciations the ice occupied the same Mangatepopo Valley. The ice in this valley may not even have entirely disappeared between those two glaciations because the temperature never got as warm as it is today.
Since the last glaciation lava has finally been able to flow in the valley bottom. The hut is on Ngauruhoe lava that is more than about 1800 years old (we know this because the pumice from the most recent Taupo Eruption is on top). And to your east you can see the flow front of a younger Ngauruhoe lava that is on top of Taupo Eruption pumice (the track deviates there to the north around the younger flow front).