We investigate the laminar-to-turbulent transition for non-Newtonian Herschel–Bulkley fluids that exhibit either a shear-thinning or shear-thickening behavior. The reduced-order model developed in this study also includes the effect of yield-stress for the fluid. Within our model framework, we investigate how the Newtonian dynamics change when significant non-Newtonian effects are considered either via the flow index n or the yield-stress τ0 or both. We find that an increase in τ0 as well as a decrease in n lead to a delayed transition if a perturbation of the given turbulent intensity is injected at various radial locations. As the radial position of the injection for the perturbation is varied in this study, our reduced-order model allows for the investigation of the flow receptivity to the finite-amplitude perturbations and to their radial position of inception. We observe that, for a given mean flow profile, the same perturbation becomes more prone to induce turbulence the closer it approaches the wall because of its initial amplitude being relatively higher with respect to the local mean flow. An opposite trend is found when the perturbation amplitude is rescaled on the local mean flow.