The efficiency of conversion of electrical power into fluidic power in an electrohydrodynamic (EHD) pump depends on the bulk fluid velocity. An analytical formulation is developed for calculation of the efficiency of an EHD pump, with and without the presence of a superimposed flow due to an externally imposed pressure gradient. This formulation is implemented into a numerical model, which is used to investigate the effect of bulk fluid velocity on the efficiency of the EHD action. In particular, the net flow due to the combined action of EHD and a positive or negative external pressure gradient is computed. Both ion-drag pumps and induction EHD pumps are considered. Pumps based on the ion-drag principle that are studied include a one-dimensional pump, a two-dimensional pump driven by a stationary potential gradient, and another driven by a traveling potential wave. Two-dimensional repulsion-type and attraction-type induction pumping caused by a gradual variation in the electrical conductivity of the fluid is also investigated. The efficiency of EHD pumps exhibited a strong dependence on bulk fluid velocity: for the two-dimensional steady ion-drag pump, for example, the efficiency increased from less than 2% to 22% under the influence of an external pressure gradient. The corresponding increase in efficiency for a two-dimensional repulsion-type EHD pump was from 0.26% to 24.5%.