In sparse signal recovery, to overcome the ‐norm sparse regularisation's disadvantages tendency of uniformly penalise the signal amplitude and underestimate the high‐amplitude components, a new algorithm based on a non‐convex minimax‐concave penalty is proposed, which can approximate the ‐norm more accurately. Moreover, the authors employ the ‐norm loss function instead of the ‐norm for the residual error, as the ‐loss is less sensitive to the outliers in the measurements. To rise to the challenges introduced by the non‐convex non‐smooth problem, they first employ a smoothed strategy to approximate the ‐norm loss function, and then use the difference‐of‐convex algorithm framework to solve the non‐convex problem. They also show that any cluster point of the sequence generated by the proposed algorithm converges to a stationary point. The simulation result demonstrates the authors’ conclusions and indicates that the algorithm proposed in this study can obviously improve the reconstruction quality.