  We establish a Freidlin-Wentzell type large deviation principle (LDP) for a
class of stochastic partial differential equations with locally monotone
coefficients driven by L\'evy noise. Our results essentially improve a recent
work on this topic (Bernoulli, 2018) by the second named author of this paper
and his collaborator, because we drop the compactness embedding assumptions,
and we also make the conditions for the coefficient of the noise term more
specific and weaker. To obtain our results, we utilize an improved sufficient
criteria of Budhiraja, Chen, Dupuis, and Maroulas for functions of Poisson
random measures, and the techniques introduced by the first and second named
authors of this paper in \cite{WZSIAM} play important roles.
  As an application, for the first time, the Freidlin-Wentzell type LDPs for
many SPDEs driven by L\'evy noise in unbounded domains of $\mathbb{R}^d$, which
are generally lack of compactness embeddings properties, are achieved, like
e.g., stochastic $p$-Laplace equation, stochastic Burgers-type equations,
stochastic 2D Navier-Stokes equations, stochastic equations of non-Newtonian
fluids, etc.
