A turboprop operates on the same principle as a turbofan only the prop acts as the primary air compressor instead of a big bladed fan. The worry is that the ash contains rock and glass so with the intense heat of the engine liquifying the particles that the intakes on the 'can' portion of the combustion chamber get smothered, or the turbines themselves are destroyed as the particulates solidify on exit. That and the external structure of the aircraft is sandblasted. I think even though it may be safe to fly, it's not good for the longevity of the engine, and engine pollutants might never be discovered until it's too late.
Which flight school did you sign up to? I'd be surprised they'd admit you at all without sucessfully acquiring the finances even if you passed selection.
