For example, when I want to use a small turbojet or low bypass turbofan as both the APU and bleed air supplier for blown flaps, ailerons, elevators, (like the ShinMaywa US-2 amphibious plane) and rudders, (and hot wing deicing) there's a lot of internal ducting, and a large number of them are quite narrow, so having soot in them is like having thrombus in capillaries, so it's best that it doesn't contain so much as a single carbon atom in it.
EDIT: OK, the small engine mentioned above is used as a gas generator, not for thrust, that's the main engines' job. For the sake of the argument, let us assume the main engines are not available for bleed air like in real-life large airliners because they are say mounted on gimbals in a tilt-turboprop. If this sounds artificial, how about having a complete set of blown surfaces so the tilt-turboprop can have control authorities in pitch, roll, and yaw at airspeed 0 knots, i.e. when it's taking off vertically. Being a tilt-turboprop means it can not have helicopter-like variable pitch propellers like the V22 Osprey and has to rely on external measures for attitude control.
EDIT: It doesn't matter how weak the alternative fuel is as long as it can support the small jet engine's N1 RPM. It is only for the small engine.