In this paper, we consider a broad class of nonconvex and nonsmooth composition optimization problems that can be used to model many applications in signal processing and image processing, such as sparse signal recovery and image restoration. However, due to the nonconvex nonsmooth properties of the objective function, solving this class of problems using classical methods like alternating minimization will face challenges in theoretical analysis and numerical calculation. For this, we propose a proximal alternating partially linearized minimization (PAPLM) algorithm by linearizing the nonconvex term and combining it with the traditional proximal algorithm. This algorithm enjoys simple and well-defined updates. By leveraging the Kurdyka-Łojasiewicz property, we prove that any sequence generated by the PAPLM algorithm globally converges to a critical point of the objective function under weaker assumptions. Numerical experiments on perturbed compressed sensing problems suggest that the proposed algorithm can achieve superior performance.