In this paper, a sparse signal recovery algorithm using Bayesian linear regression with Cauchy prior (BLRC) is proposed. Utilizing an approximate expectation maximization(AEM) scheme, a systematic hyper-parameter updating strategy is developed to make BLRC practical in highly dynamic scenarios. Remarkably, with a more compact latent space, BLRC not only possesses essential features of the well-known sparse Bayesian learning (SBL) and iterative reweighted l2 (IR-l2) algorithms but also outperforms them. Using sparse array (SPA) and coprime array (CPA), numerical analyses are first performed to show the superior performance of BLRC under various noise levels, array sizes, and sparsity levels. Applications of BLRC to sparse multiple-input and multiple-output (MIMO) radar array signal processing are then carried out to show that the proposed BLRC can efficiently produce high-resolution images of the targets.