Plasma devices generating a sequence of streamers over an inert gas jet in ambient air have been widely used recently in anticancer therapy. An enhancement of electric field carried by the streamers to the target surface is a way to intensify the plasma jet impact on a cell life cycle. In this paper, an increase of the surface electric field without a rise of discharge power is provided with an external biased electrode installed between the plasma device and target. The effect of the presence of this biased electrode on cathode-directed streamer dynamics and its interaction with conductive or charged dielectric surfaces are studied in 2D fluid model simulations and in the experiment. Two stages of the ionization process induced by the streamer near the plasma–target interface are observed in the experiment and simulations. The first stage is related to the streamer approaching the surface. The second stage is the stationary self-sustained ionization and lasts much longer during a positive cycle of sinusoidal voltage applied to the electrode inside of the plasma device. A much higher surface electric field is registered on the dielectric surface compared to the conductive one, due to an accumulation of the surface charge, especially with the negative bias on the external electrode. An area on the dielectric surface exposed by the plasma jet is much larger compared to the conductive one, due to spreading the surface ionization wave over the dielectric.