The two-stage electron acceleration/heating model (Wu et al 2017 Nucl. Fusion57 016007 and Wu et al 2016 Phys. Plasmas23 123116) is extended to the study of laser magnetized-plasmas interactions at relativistic intensities and in the presence of large-scale preformed plasmas. It is shown that the electron-heating efficiency is a controllable value by the external magnetic fields. Detailed studies indicate that for a right-hand circularly polarized laser, the electron heating efficiency depends on both strength and directions of external magnetic fields. The electron-heating is dramatically enhanced when the external magnetic field is of . When magnetic field is of negative direction, i.e. , it trends to suppress the electron heating. The underlining physics—the dependences of electron-heating on both the strength and directions of the external magnetic fields—is uncovered. With , the electron-heating is explained by the synergetic effects by longitudinal charge separation electric field and the reflected 'envelop-modulated' CP laser. It is indicated that the 'modulation depth' of reflected CP laser is significantly determined by the external magnetic fields, which will in turn influence the efficiency of the electron-heating. While with , a laser front sharpening mechanism is identified at relativistic laser magnetized-plasmas interactions, which is responsible for the dramatical enhancement of electron-heating.