  We study a reaction-diffusion evolution equation perturbed by a space-time
L\'evy noise. The associated Kolmogorov operator is the sum of the
infinitesimal generator of a $C_0$-semigroup of strictly negative type acting
in a Hilbert space and a nonlinear term which has at most polynomial growth, is
non necessarily Lipschitz and is such that the whole system is dissipative.
  The corresponding It\^o stochastic equation describes a process on a Hilbert
space with dissi- pative nonlinear, non globally Lipschitz drift and a L\'evy
noise. Under smoothness assumptions on the non-linearity, asymptotics to all
orders in a small parameter in front of the noise are given, with detailed
estimates on the remainders.
  Applications to nonlinear SPDEs with a linear term in the drift given by a
Laplacian in a bounded domain are included. As a particular case we provide the
small noise asymptotic expansions for the SPDE equations of FitzHugh Nagumo
type in neurobiology with external impulsive noise.
