This paper centers on stochastic Hopfield neural networks with variable coefficients and infinite delay. First, we propose an integral inequality that improves and extends some existing works. Second, by employing some inequalities and stochastic analysis techniques, some sufficient conditions for ensuring $p$th moment generalized exponential stability are established. Our results do not necessitate the construction of a complex Lyapunov function or rely on the assumption of bounded variable coefficients, and our results expand some existing works. At last, to illustrate the efficacy of our result, we present several simulation examples.