We present the first experiments on electrohydrodynamic convection with a superposed Couette shear in freely suspended annular smectic C* films. Similar experiments on smectic A films have been carried out previously. In those experiments, surface-charge driven electroconvection in combination with shear leads to a pattern of travelling convective vortices. Travelling vortex patterns are also observed in smectic C* films. In contrast to smectic A films, however, the flow field in C* films is coupled to c-director orientation effects that allow a direct visualization of the periodic flow pattern. The critical voltage for the onset of convection was determined by electric current measurements and compared with corresponding results for smectic A films. Measurements at various shear rates show that the onset is suppressed by the shear.