  We report x-ray total scattering and pair distribution function (PDF) studies
of the structural relaxation around interstitial manganese (Mn$_i$) in
ferromagnetic Mn$_{1+\delta}$Sb ($0.03 \le \delta \le 0.23$) alloys, guided by
density functional theory (DFT). Refinements to the experimental PDF using a
crystallographically constrained structural model indicate an expansion in the
equatorial plane of the Mn$_i$Sb$_5$ trigonal bipyramidal site, which
introduces significant positional disorder in addition to the nominally-random
occupation of interstitial voids. Observation of a weak diffuse signal near the
symmetry-forbidden (001) reflection position is indicative of correlated
disorder from the clustering of Mn$_i$. Density functional relaxation of
supercells approximating the $\delta = 0.08$, $0.15,$ and $0.23$ compositions
provides improved models that accurately describe the short-range structural
distortions captured in the PDFs. Such structural relaxation increases the DFT
calculated moment on Mn$_i$, which aligns antiparallel to the primary Mn
moments, but leads to insubstantial changes in the average Mn and Sb moments
and moments of Mn and Sb proximal to interstitials, thus providing a more
accurate description of the observed bulk magnetic properties.
