We use spin dynamics simulations to determine the origin of the unusual correlated diffuse scattering, characterised by half-moon shapes bridging the magnetic Bragg peaks, observed in the polarised elastic neutron scattering from manganese tungstate, MnWO₄. We first fit a Heisenberg Hamiltonian with twelve nearest-neighbour exchange interactions and single-ion anisotropy to the experimental ground-state magnon dispersion. We then show via spin dynamics simulations that our model Hamiltonian both reproduces the experimentally observed half-moon features and captures their persistence into the paramagnetic regime. Moreover, we identify the three-dimensional, competing antiferromagnetic interactions driving this behavior. Our work complements earlier studies of half-moon-shaped signatures in pyrochlore and triangular structures, by providing insight into their origin in a zigzag chain geometry with three-dimensional competing exchange interactions.