We experimentally study the dynamics of the generation of multiple sidebands by means of a quasi-phase-matched four-wave mixing (FWM) process occurring in a dispersion-oscillating, highly nonlinear optical fiber. The fiber under test is pumped by a ns microchip laser operating in the normal average group-velocity dispersion regime and in the telecom C band. We reveal that the growth of higher-order sidebands is strongly influenced by the competition with cascade FWM between the pump and the first-order quasi-phase matched sidebands. The properties of these competing FWM processes are substantially affected when a partially coherent pump source is used, leading to a drastic reduction of the average power needed for sideband generation.