A parametric study of the impingement of a helium kHz atmospheric pressure plasma jet on a flat glass surface was performed by means of time-resolved intensified charge-coupled device imaging. The development of the plasma on the target is linked to the plasma evolution in the source and governed by the power supply. The glass surface takes part in the elongation of the plasma jet by the virtue of two mechanisms: the local enhancement of the electric field and the supply of pre-deposited charge. The evidence for the pre-deposited charge is the formation of a sheath on the glass surface, and the faint discharge formed on the glass surface during the negative voltage slope starting at the maximum of the negative current peak. The influence of the gas flow dynamics taking into account various gas flows, incident angles and distances is more important for the behaviour of the discharge on the surface than the voltage amplitude or the geometry of the source. The capacitance of the target strongly modifies the interaction with the plasma jet and increases the deposited surface charge density, featuring a streamer-like propagation mechanism in the case of high electric field enhancement at the surface.