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We establish a Freidlin-Wentzell type large deviation principle (LDP) for aclass of stochastic partial differential equations with locally monotonecoefficients driven by L\'evy noise. Our results essentially improve a recentwork on this topic (Bernoulli, 2018) by the second named author of this paperand his collaborator, because we drop the compactness embedding assumptions,and we also make the conditions for the coefficient of the noise term morespecific and weaker. To obtain our results, we utilize an improved sufficientcriteria of Budhiraja, Chen, Dupuis, and Maroulas for functions of Poissonrandom measures, and the techniques introduced by the first and second namedauthors of this paper in \cite{WZSIAM} play important roles.  As an application, for the first time, the Freidlin-Wentzell type LDPs formany SPDEs driven by L\'evy noise in unbounded domains of $\mathbb{R}^d$, whichare generally lack of compactness embeddings properties, are achieved, likee.g., stochastic $p$-Laplace equation, stochastic Burgers-type equations,stochastic 2D Navier-Stokes equations, stochastic equations of non-Newtonianfluids, etc.

Large deviations for locally monotone stochastic partial differential equations driven by L\'evy noise