  The asymptotic analysis of a class of stochastic partial differential
equations (SPDEs) with fully locally monotone coefficients covering a large
variety of physical systems, a wide class of quasilinear SPDEs and a good
number of fluid dynamic models is carried out in this work. The aim of this
work is to develop the large deviation theory for small Gaussian as well as
Poisson noise perturbations of the above class of SPDEs. We establish a
Wentzell-Freidlin type large deviation principle for the strong solutions to
such SPDEs perturbed by L\'evy noise in a suitable Polish space using a
variational representation (based on a weak convergence approach) for
nonnegative functionals of general Poisson random measures and Brownian
motions. The well-posedness of an associated deterministic control problem is
established by exploiting pseudo-monotonicity arguments and the stochastic
counterpart is obtained by an application of Girsanov's theorem.
