We investigate the near-field radiative heat transfer between a normally and/or laterally shifted nanoparticle and a planar fused silica slab coated with a strip graphene grating. For this study we develop and use a scattering matrix approach derived from Fourier modal method augmented with local basis functions. We find that adding a graphene sheet coating on the slab can already enhance the heat flux by about 85%. We show that by patterning the graphene sheet coating into a grating, the heat flux is further increased, and this happens thanks to the a topological transition of the plasmonic modes from circular to hyperbolic one, which allows for more energy transfer. The lateral shift affects the accessible range of high-$k$ modes and thus affects the heat flux, too. By moving the nanoparticle laterally above the graphene grating, we can obtain an optimal heat flux with strong chemical potential dependance above the strips. For a fixed graphene grating period ($D=1μ$m) and not too large normal shift (separation $d<800$nm), two different types of lateral shift effects (e.g., enhancement and inhibition) on heat transfer have been observed. As the separation $d$ is further increased, the lateral shift effect becomes less important. We show that the lateral shift effect is sensitive to the geometric factor $d/D$. Two distinct asymptotic regimes are proposed: (1) the inhibition regime ($d/D<0.85$), where the lateral shift reduces the heat transfer, and (2) the neutral regime ($d/D \geq 0.85$) where the effect of the lateral shift is negligible. In general, we can say that the geometric factor $d/D \approx 0.85$ is a critical point for the lateral shift effect. Our predictions can have relevant implications to the radiative heat transfer and energy management at the nano/micro scale.