<p>Deterministic row-action methods, like the maximal weighted residual Kaczmarz (MWRK) method and its momentum-accelerated variant (mMWRK), are known to converge linearly for consistent linear systems. However, their behavior for inconsistent problems—even those arising from a noisy right-hand side—is markedly different, typically exhibiting semiconvergence. This work provides a rigorous analysis of this phenomenon. We first investigate the semiconvergence properties of MWRK and mMWRK when applied to inconsistent problems. Building on this analysis, we propose two augmented solvers designed to overcome the semiconvergence and converge to the least-squares solution. We prove that these augmented methods achieve a linear convergence rate. Numerical experiments on classical inconsistent problems substantiate our theoretical findings and demonstrate the efficacy of the proposed augmented solvers.</p>