We present a compressive sensing based defect detection by multiple input multiple output (MIMO) wireless radar. Here, defects are inside a layered material structure, therefore, due to reflections from the surface of the layered material structure the defect detection is challenging. By utilizing a low-rank nature of the reflections of the layered material structure and sparse nature of the defects, we propose a method based on rank minimization and sparse recovery. To improve the accuracy in the recovery of low-rank and sparse components, we propose a non-convex approach based on the iteratively reweighted nuclear norm and iteratively reweighted $\ell_1-$norm algorithm. Our numerical results show that the proposed method is able to demix and recover the signalling responses of the defects and layered structure successfully from substantially reduced number of observations. Further, the proposed approach outperforms the state-of-the-art clutter reduction approaches