Abstract: In this paper, we introduce a concept of Poisson p-mean almost automorphy for stochastic processes and give the composition theorems for (Poisson) p-mean almost automorphic functions under non-Lipschitz conditions. Our abstract results are, subsequently, applied to study a class of neutral stochastic evolution equations driven by L\'evy noise, and we present sufficient conditions for the existence of square-mean almost automorphic mild solutions. An example is provided to illustrate the effectiveness of the proposed result.