We introduce a minimal effective framework designed to address the anomalous low-temperature thermodynamics observed in strongly disordered interacting systems near a metal-insulator transition, such as doped semiconductors. Our approach examines how a strongly inhomogeneous electronic bath quenches dilute pairs of local magnetic moments. By solving the resulting two-impurity Kondo-Heisenberg problem within a large-$𝒩$ variational framework, we capture the competition between the Kondo screening and random-singlet formation. Both mechanisms give rise to singular thermodynamic responses, producing a magnetic susceptibility of the form $\chi (T)\sim T^{-\alpha }$, where the exponent $\alpha$ evolves smoothly with the model parameters and saturates in the metallic regime, consistent with experimental observations. Our results establish a unified framework that connects random-singlet physics and disorder-induced Kondo screening in strongly disordered correlated materials.