Flow over aligned and staggered cube arrays is a classic model problem for rough-wall turbulent boundary layers. Earlier studies of this model problem mainly looked at rough surfaces with a moderate coverage density, i.e.  $\unicodeSTIX{x1D706}_{p}>O(3\,\%)$ , where $\unicodeSTIX{x1D706}_{p}$ is the surface coverage density and is defined to be the ratio between the area occupied by the roughness and the total ground area. At lower surface coverage densities, i.e.  $\unicodeSTIX{x1D706}_{p}<O(3\,\%)$ , it is conventionally thought that cubical roughness acts like isolated roughness elements; and that the single-cube drag coefficient, i.e.  $C_{d}\equiv f/(\unicodeSTIX{x1D70C}U_{h}^{2}h^{2})$ , equals $C_{R}$ . Here, $f$ is the drag force on one cubical roughness element, $\unicodeSTIX{x1D70C}=\text{const.}$ is the fluid density, $h$ is the height of the cube, $U_{h}$ is the spatially and temporally averaged wind speed at the cube height, and $C_{R}$ is the drag coefficient of an isolated cube. In this work, we conduct large-eddy simulations and direct numerical simulations of flow over wall-mounted cubes with very low surface coverage densities, i.e.  $0.08\,\%<\unicodeSTIX{x1D706}_{p}<4.4\,\%$ . The large-eddy simulations are at nominally infinite Reynolds numbers. The results challenge the conventional thinking, and we show that, at very low surface coverage densities, the single-cube drag coefficient may increase as a function of $\unicodeSTIX{x1D706}_{p}$ . Our analysis suggests that this behaviour may be attributed to secondary turbulent flows. Secondary turbulent flows are often found above spanwise-heterogeneous roughness. Although the roughness considered in this work is nominally homogeneous, the secondary flows in our simulations are very similar to those observed above spanwise-heterogeneous surface roughness. These secondary vortices redistribute the fluid momentum in the outer layer, leading to high-momentum pathways above the wall-mounted cubes and low-momentum pathways at the two sides of the wall-mounted cubes. As a result, the spatially and temporally averaged wind speed at the cube height, i.e.  $U_{h}$ , is an underestimate of the incoming flow to the cubes, which in turn leads to a large drag coefficient  $C_{d}$ .