Superconductivity in disordered locally noncentrosymmetric materials: An application to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>CeRh</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>As</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>

Layered three-dimensional centrosymmetric crystals can exhibit characteristics of noncentrosymmetric materials. This happens when each individual layer alone lacks inversion, but, combined, inversion symmetry is restored; hence the designation: locally noncentrosymmertic superconductors (LNCSs). In LNCSs, effects impurities and subdominant magnetic field induced pairing channels remain unexplored. Using a minimal model, we examine all show that there always superconducting instability favored at high fields, which substantially alter -- temperature phase diagram. Also, find diagram responds to disorder in non-monotonic way, be subjected experimental verification. We apply these ideas recently unveiled two-phase CeRh$_2$As$_2$. identify two phases as singlet-triplet mixed even- odd-parity states low respectively. Furthermore, predict presence also for in-plane fields cleaner samples, since high-field could have been so far hindered by impurity effects.