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 Lévy 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.