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We write down an $\mathcal{N}=1$ supersymmetric extension for non-Abeliangauge theories in (1+3) dimensions with a Lorentz- and CPT-violating term ofthe Carroll-Field-Jackiw type. By including effects of the background(supersymmetric) fermion bilinears that accompany Lorentz-symmetry violation ina Carroll-Field-Jackiw scenario, we investigate both the gauge boson andgaugino dispersion relations in order to compute their respective masses interms of the background structures. Such results open up a potential pathtowards a possible mechanism for gaugino-gauge boson conversion, an analogue ofthe Primakoff effect, induced here not by an external magnetic field, butinstead by the (Majorana) fermionic sector of the supersymmetry multiplet inthe backstage of the Lorentz-symmetry violation.

Aspects of the gauge boson-gaugino mixing in a supersymmetric scenario with Lorentz-symmetry violation