Sparse recovery (SR) based on spiking neural networks has been shown to be computationally efficient with ultra-low power consumption. However, existing spiking-based sparse recovery (SSR) algorithms are designed for the convex 1-norm regularized SR problem, which often underestimates the true solution. This paper proposes an adaptive version of SSR, i.e., A-SSR, to optimize a class of non-convex regularized SR problems and analyze its global asymptotic convergence. The superiority of A-SSR is validated with synthetic simulations and real applications, including image reconstruction and face recognition. Furthermore, it is shown that the proposed A-SSR essentially improves the recovery accuracy by avoiding systematic underestimation and obtains over 4 dB PSNR improvement in image reconstruction quality and around 5% improvement in recognition confidence. At the same time, the proposed A-SSR maintains energy efficiency in hardware implementation. When implemented on the neuromorphic Loihi chip, our method consumes only about 1% of the power of the iterative solver FISTA, enabling applications under energy-constrained scenarios.