Yeah but just like cars in the desert you risk the dust filter getting totally clogged by the dust. With a car, you can stop and simply unclog the filter by hand - impossible on a piston engine in flight.
Sure, there are at least two ways, but you won't get it fuel-efficient enough for commercial service. You could do research flights into the volcanic ash.
One choice is a low-compression jet engine that doesn't burn hot. Temperatures need to stay lower than the melting point of volcanic ash. Add a durable coating (maybe: sapphire, diamond, tungsten carbide...) to resist abrasion and you're all set.
Another choice is to use an engine that is part turboprop and part rocket. The turbine portion is fed entirely from internal stores, for example liquid oxygen and liquid methane. Keep them balanced for maximum power, or dilute/unbalance them to avoid the need for exotic materials. Ash never enters the turbine. To get back a little efficiency, you put a prop on the front instead of just having a rocket.
You'll want to also put a sapphire coating on the windshield. Treating the wing leading edges could also be a good idea; maybe for that you could consider titanium-aluminum-nitride.
But it still does seem like it could be dealt with - the abrasion does not completely destroy it, so it seems just incremental hardness improvement would handle it.
The damaged blades in the photo are turbine inlet guide vanes. They are attached to the static part of the engine in front fo the rotating turbine and are the first thing the hot (1500C) gasses from combustion hit. To prevent them from melting the blades are hollow and compressor bleed air (300C) flows through them and out the holes visible on the blades. This cooling air forms an insulating layer to protect the blade surfaces. Additionally they will have a ceramic thermal barrier coating. The actual turbine blades are similar except that they have to cope with tremendous loads from centrifugal force and the power transfer from the the gas stream to the shaft.
The damage shown indicates loss of the thermal barrier coating at the leading edge and deformation of the leading edge and the cooling holes. The precise shapes are critical to maintaining the cool air film on the blade. Basically these vanes are toast and in a short time would be burnt melted toast.
Let me recommend the book: “The Jet Engine” published by Rolls-Royce. Jet engines are pretty amazing. They seem simple in principle, but the optimizations needed to make a good one are beyond most countries and companies capabilities.
True, but ablative tips would likely mean engine overhauls at increased frequency, significantly increasing cost. At that cost vs benefit, likely better to just go back to piston and prop.
Does anyone here have a perspective on the engineering that would be required to build ash-proof engines? Perhaps similarly, drone/bird-proof engines?