Time-reversal symmetry-breaking Weyl semimetals (WSMs) exhibit an interesting optical response in the mid-infrared band, offering what we believe to be a novel approach to study the manipulation of the photonic spin Hall effect (PSHE). This article theoretically investigates the PSHE phenomenon in a prism-coupled structure composed of the WSMs. It is found that the excitation of surface plasmon polaritons (SPPs) and the nonreciprocity inherent in WSMs have a positive effect on enhancing the PSHE phenomenon of the reflected light. Especially, the PSHE can also be manipulated by varying the separation distance and twist angle between the Weyl nodes. Through parameter optimization, we obtained a lateral displacement of 12.14 μm. The effects of the separation distance between Weyl nodes, the twist angle, and the incident light angle on the PSHE were further elucidated. We believe the tunable PSHE with WSMs can provide promising avenues for developing spintronic devices, as well as other photonic applications.