This article investigates finite-time stability (FTS) and finite-time contractive stability (FTCS) of discontinuous nonlinear fractional-order (FO) systems with time-delay and state-dependent delayed impulses. Lyapunov–Razumikhin (LR) conditions and impulse perturbations yield the essential and adequate conditions for stability criteria. Based on the main concept of this work, we investigate the stability analysis of retarded FO neural networks (NNs) with time delays, FO-delayed Cohen–Grossberg NNs, and FO-delayed bidirectional associative memory NNs within the framework of the Filippov map due to the fact that the neuron activation functions are discontinuous. The above NNs will verify the Lyapunov–Razumikhin conditions, and finally, three numerical simulations are provided to demonstrate the efficacy of this framework.