A new path equation in absolute parallelism (AP) geometry is derived. The equation is a generalization of three path equations derived in a previous work. It can be considered as a geodesic equation modified by a torsion term, whose numerical coefficient jumps by steps of one half. The torsion term is parametrized using the fine structure constant. It is suggested that the new equation may describe the trajectories of spinning particles under the influence of a gravitational field, and the torsion term represents a type of interaction between the quantum spin of the moving particle and the background field. Weak field limits of the new path equation show that the gravitational potential felt by a spinning particle is different from that felt by a spinless particle (or a macroscopic body). As a byproduct, and in order to derive the new path equation, the AP-space is reconstructed using a new affine connexion preserving metricity. The new AP-structure has non-vanishing curvature. In certain limits, the new AP-structure can be reduced either to the ordinary Riemannian space, or to the conventional AP-space.