The corkscrew effect is irrelevant on twin engined airplanes.
Also, the torque produced by the Newton' s 3rd law itself only affects single-engine airplanes
True when both screws are turning, but the original question was regarding the causes of yaw when one engine fails (besides the obvious asymmetrical thrust).
That's one of the best explanations of P-Factor I've ever seen, but in this instance, we are discussing specifically the stage of the takeoff when the angle of the propeller disc axis and the relative airflow is quite near to perpendicular. As you mentioned, P-Factor becomes a primary controllability issue once AoA is increased and liftoff is achieved. While still on the ground roll, however, the factors I listed (torque, propwash and x-wind) will have a larger effect than P-Factor on a twin engine aircraft that has suddenly and unexpectedly become a single.