High-throughput density functional calculations are used to investigate the effect of interstitial B, C and N atoms on 21 alloys reported to crystallize in the cubic Cu$_3$Au structure. It is shown that the interstitials can have a significant impact on the magneto-crystalline anisotropy energy (MAE), the thermodynamic stability and the magnetic ground state structure, making these alloys interesting for hard magnetic, magnetocaloric and other applications. For 29 alloy/interstitial combinations the formation of stable alloys with interstitial concentrations above 5% is expected. In Ni$_3$Mn interstitial N induces a tetragonal distortion with substantial uniaxial MAE for realistic N concentrations. Mn$_3X$N$_x$ ($X$=Rh, Ir, Pt and Sb) are identified as alloys with strong magneto-crystalline anisotropy. For Mn$_3$Ir we find a strong enhancement of the MAE upon N alloying in the most stable collinear ferrimagnetic state as well as in the non-collinear magnetic ground state. Mn$_3$Ir and Mn$_3$IrN show also interesting topological transport properties. The effect of N concentration and strain on the magnetic properties are discussed. Further, the huge impact of N on the MAE of Mn$_3$Ir and a possible impact of interstitial N on amorphous Mn$_3$Ir, a material that is indispensable in todays data storage devices, are discussed at hand of the electronic structure. For Mn$_3$Sb, non-collinear, ferrimagnetic and ferromagnetic states are very close in energy, making this material potentially interesting for magnetocaloric applications. For the investigated Mn alloys and competing phases, the determination of the magnetic ground state is essential for a reliable prediction of the phase stability.
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