Given the measurement matrix $A$ and the observation signal $y,$ the central purpose of compressed sensing is to find the most sparse solution of the underdetermined linear system $y = Ax+z,$ where $x$ is the $s$-sparse signal to be recovered and $z$ is the noise vector. Zhou and Yu [Front. Appl. Math. Stat., 5 (2019), Article 14] recently proposed a novel non-convex weighted $ℓ_r − ℓ_1$ minimization method for effective sparse recovery. In this paper, under newly coherence-based conditions, we study the non-convex weighted $ℓ_r − ℓ_1$ minimization in reconstructing sparse signals that are contaminated by different noises. Concretely, the results reveal that if the coherence $\mu$ of measurement matrix $A$ fulfills $$\mu < \kappa(s; r, α, N), \ s > 1, α ^{\frac{1}{ r}} N ^{\frac{1}{ 2}} < 1,$$ then any $s$-sparse signals in the noisy scenarios could be ensured to be reconstructed robustly by solving weighted $ℓ_r − ℓ_1$ minimization non-convex optimization problem. Furthermore, some central remarks are presented to clear that the reconstruction assurance is much weaker than the existing ones. To the best of our knowledge, this is the first mutual coherence-based sufficient condition for such approach.