Recovering signals, from their linear measurements, that are sparse and nonnegative plays a pivotal role in numerous applications. Recent advancements in model-based deep learning, particularly deep unrolling techniques, have emerged as powerful alternatives to traditional iterative solvers, offering a balance between interpretability and computational efficiency. In this paper, we introduce a novel unrolled network architecture inspired by the ReLU-based hard thresholding (RHT) algorithm, tailored for sparse and nonnegative signal recovery. An advantage of this unrolling approach lies in providing a faster inference. Unlike conventional hard thresholding used in RHT, we incorporate a differentiable hard thresholding operator as an activation function within the unrolled iterations, which yields improved gradient flow and learning dynamics during training. The resulting framework, termed learned ReLU-based hard thresholding (LRHT), preserves the structural advantages of RHT while enhancing adaptability through learnable parameters. From the experiments we show that the LRHT network shows effective reconstruction results for solving the stated inverse problem.