What is the criterion for an addition that invalidates the prayer, and is the prayer invalidated by repeating part of a pillar (such as the nose in prostration or the head in the taslim) to correct a dhikr or complete a tashahhud?

The prayer is not invalidated by lifting the nose while the forehead remains on the ground, because the prostration is essentially (validated) by the forehead. And putting the nose back on the ground is not considered an addition. Likewise, a slight turning of the face is reprehensible if not for a need, but it does not invalidate the prayer.