In order to solve the problem of insufficient torque for flux weakening in permanent magnet machines, a novel flux-torque regulation hybrid excitation machine with axial-radial magnetic circuit (FTRHEM) is proposed. To solve the problem that the cogging torque of the proposed machine is caused by multiple factors and the coupling relationship is complicated, an analytical expression for the cogging torque is derived first. Then, the influence of the axial excitation device on the cogging torque is determined. The influence mechanism of the relative positions between the axial excitation device and stator on the cogging torque is revealed. A dislocation method is proposed to reduce the cogging torque, and its effectiveness is verified by the finite element analysis (FEA). Furthermore, the reduction of the cogging torque in FTRHEM with different pole-slot combinations by this method is comprised and analyzed, which demonstrates the feasibility of the method. Finally, the FEA results are combined with experiments to verify the accuracy of simulations.