A practical finite-time neural pseudo inverse control (PFTNPIC) strategy is proposed to address the finite-time tracking control of fractional-order systems subject to asymmetric hysteresis and time-delay. To compensate for the asymmetric hysteresis effect, a novel pseudo inverse algorithm is developed based on the asymmetric shifted Prandtl–Ishlinskii model, thereby avoiding the need to construct an explicit hysteresis inverse model. Furthermore, a framework that integrates a radial basis function neural network with the finite cover lemma is employed to eliminate the time-delay issue. With the proposed PFTNPIC approach, all signals in the closed-loop system are shown to be semiglobally uniformly ultimately bounded and the tracking error is guaranteed to converge to a small neighborhood of origin within finite time. Finally, the effectiveness of the proposed strategy is validated through experiments conducted on a displacement platform driven by a giant magnetostrictive actuator.