To clarify the ultrafast temporal interplay of the different photocurrent mechanisms occurring in single InAs-nanowire-based circuits, an on-chip photocurrent pump-probe spectroscopy based on coplanar striplines was utilized. The data are interpreted in terms of a photo-thermoelectric current and the transport of photogenerated holes to the electrodes as the dominating ultrafast photocurrent contributions. Moreover, it is shown that THz radiation is generated in the optically excited InAs-nanowires, which is interpreted in terms of a dominating photo-Dember effect. The results are relevant for nanowire-based optoelectronic and photovoltaic applications as well as for the design of nanowire-based THz sources.