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.
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