A distinctly temperature-dependent Nernst coefficient, ν, which is strongly enhanced over that of LaCu(2)Si(2), is observed between T=2 and 300 K for CeCu(2)Si(2) and Ce(0.8)La(0.2)Cu(2)Si(2). The enhanced ν(T) is determined by the asymmetry of the on-site Kondo (conduction electron -4f electron) scattering rate. Taking into account the measured Hall mobility, μ(H), the highly unusual thermopower, S, of these systems can be semiquantitatively described by S(T)=-ν(T)/μ(H)(T), which explicitly demonstrates that the thermopower originates from the local Kondo scattering process over a wide temperature range from far above to well below the coherence temperature (≈20 K for CeCu(2)Si(2)). Our results suggest that the Nernst effect can act as a proper probe of local charge-carrier scattering. This promises to impact on exploring the unconventional enhancement of the thermopower in correlated materials suited for potential applications.