The electron dynamics and the Thomson backscattering of an electron moving in a combined field of a tightly focused Gaussian laser pulse and an external uniform magnetic field are investigated in detail. It is found that by considering the tightly focused Gaussian laser pulse, the electron can be pushed out from the laser pulse by the ponderomotive force, resulting in the symmetry breaking of the electron dynamics from the Gaussian envelope, which can dramatically enhance the radiation intensity. It is also found that by introducing an external magnetic field, the emergence of the cyclotron motion of the electron under the external magnetic field also breaks the symmetry of the electron dynamics and enhances the radiation. Especially in the resonance case, i.e., the cyclotron frequency of the electron is close to the laser frequency, the emission spectrum is further enhanced due to the great extension of the interaction time and the symmetry breaking by the beat wave between the helix motion and the cyclotron motion of the electron in the combined field, and a platform with high radiation intensity containing the THz band has also appeared.