We investigate the one-dimensional diffusion of a particle in a piecewise linear potential superimposed with a step of a harmonically modulated height. Employing the matching conditions, we solve the corresponding Fokker-Planck equation and we analyze nonlinear features of the particle's mean position as a function of time. We present detailed results in two physically relevant cases. First, we take the unperturbed potential as a symmetrical up-oriented tip, which is placed between two reflecting boundaries and we add the jump at the tip coordinate. The setting yields resonance-like behavior of the stationary-response amplitude. Second, if the discontinuity at origin is combined with the constant force in the symmetrical region between the boundaries, the stationary response displays a time-independent shift against the potential slope. The driving-induced force exhibits a resonance-like behavior both with respect to the diffusion constant and the slope of the unperturbed potential.