  We present a comprehensive study of the short-ranged nuclear and magnetic
order in the two-dimensional spin glass, Mn$_{0.5}$Fe$_{0.5}$PS$_3$. Nuclear
neutron scattering data reveal a random distribution of Mn$^{2+}$ and Fe$^{2+}$
ions within the honeycomb layers, which gives rise to a spin glass state
through inducing competition between neighbouring exchange interactions,
indicated in magnetic susceptibility data by a cusp at the glass transition,
$T_g = 35$ K. Analysis of magnetic diffuse neutron scattering data collected
for both single crystal and polycrystalline samples gives further insight into
the origin of the spin glass phase, with spin correlations revealing a mixture
of satisfied and unsatisfied correlations between magnetic moments within the
honeycomb planes, which can be explained by considering the magnetic structures
of the parent compounds, MnPS$_3$ and FePS$_3$. We found that, on approaching
$T_g$ from above, an ensemble-averaged correlation length of $\xi = 5.5(6)$
\r{A} developed between satisfied correlations, and below $T_g$, the glassy
behaviour gave rise to a distance-independent correlation between unsatisfied
moments. Correlations between the planes were found to be very weak, which
mirrored our observations of rod-like structures parallel to the c* axis in our
single crystal diffraction measurements, confirming the two-dimensional nature
of Mn$_{0.5}$Fe$_{0.5}$PS$_3$.
