Space-time adaptive processing (STAP) is critical for clutter suppression in airborne radar, but limited training data is a challenge. Sparse recovery (SR) STAP has emerged to address this, but most existing approaches are model-based and require complex parameter tuning. To overcome this, we propose a novel framework that combines model-driven and learning-based approaches using a deep convolutional neural network based on the STAP principle and the half quadratic splitting algorithm. Our approach outputs the minimum variance distortionless response spectrum for clutter suppression, resulting in superior performance and computational efficiency compared to existing SR STAP methods. Our contribution lies in the novelty of the proposed framework for combining model-driven and learning-based approaches to effectively address the challenges of STAP. Moreover, we propose a novel data mapping that guarantees the accuracy of the spectrum estimation. The simulation results confirm the effectiveness of our proposed method.