The discrete-time robust repetitive control (RC, or repetitive controller, also designated RC) problem for nonlinear systems is both challenging and practical. This paper proposes a discrete-time output-feedback RC design for a class of systems subject to measurable nonlinearities to track reference robustly with respect to the period variation. The design relies on additive state decomposition, by which the output-feedback RC problem is decomposed into an output-feedback RC problem for a linear time-invariant system and a state-feedback stabilization problem for a nonlinear system. Thanks to the decomposition, existing controller design methods in both the frequency domain and time domain can be employed to make the robustness and discretization for a nonlinear system tractable. To demonstrate the effectiveness, an illustrative example is given.