In population dynamics, cross-diffusion describes the influence of one species on the diffusion of another and, surprisingly, even though the reaction part does not present the activator-inhibitor structure, cross-diffusion terms are often the key ingredient for the appearance of spatial patterns. Furthermore, from the modelling perspective, cross-diffusion terms naturally appear in the fast-reaction limit of a "microscopic'' model (in terms of time scales) presenting only standard diffusion and fast-reaction terms, thus incorporating processes occurring on different time scales}. We exploit this technique to model the auto-toxicity effect in plant growth dynamics, i.e. negative plant-soil feedback due to the decomposed biomass of the plant on its own growth. The "macroscopic'' model presents a cross-diffusion term that allows the formation of spatial patterns without introducing water as a variable. A deeper understanding of the conditions required for non-homogeneous steady states to exist is provided by combining a detailed linear analysis with advanced numerical bifurcation methods via the continuation software \texttt{pde2path} and numerical simulations. Among the stable patterns observed, it is worth noting the presence of double-peak pulses of biomass in one-dimensional domains, which are equivalent to reinforced boundaries for the irregular spots forming on two-dimensional domains, for a wide region of parameter space.