We use DNS to study inter-scale and inter-space energy exchanges in the near-field of a turbulent wake of a square prism in terms of the KHMH equation written for a triple decomposition of the velocity field accounting for the quasi-periodic vortex shedding. Orientation-averaged terms of the KHMH are computed on the plane of the mean flow and on the geometric centreline. We consider locations between $2$ and $8$ times the width $d$ of the prism. The mean flow produces kinetic energy which feeds the vortex shedding coherent structures. In turn, these structures transfer energy to the stochastic fluctuations over all length-scales $r$ from the Taylor length $lambda$ to $d$ and dominate spatial turbulent transport of two-point stochastic turbulent fluctuations. The orientation-averaged non-linear inter-scale transfer rate $Pi^{a}$ which was found to be approximately independent of $r$ by Alves Portela et. al. (2017) in the range $lambdale r le 0.3d$ at a distance $x_{1}=2d$ from the square prism requires an inter-scale transfer contribution of coherent structures for this approximate constancy. However, the near-constancy of $Pi^a$ at $x_1=8d$ which was also found by Alves Portela et. al. (2017) is mostly due to stochastic fluctuations. Even so, the proximity of $-Pi^a$ to the turbulence dissipation rate $varepsilon$ in the range $lambdale rle d$ at $x_1=8d$ requires contributions of the coherent structures. Spatial inhomogeneity also makes a direct and distinct contribution to $Pi^a$, and the constancy of $-Pi^a/varepsilon$ close to 1 would not have been possible without it either in this near-field flow. Finally, the pressure-velocity term is also an important contributor to the KHMH, particularly at scales r larger than about $0.4d$, and appears to correlate with the purely stochastic non-linear inter-scale transfer rate when the orientation average is lifted.
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