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In this article, we consider the following class of stochastic partialdifferential equations (SPDE): \begin{equation*}\left\{\begin{aligned}\mathrm{d}\mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d}t+\mathrm{B}(t,\mathbf{X}(t))\mathrm{d}\mathrm{W}(t)+\int_{\mathrm{Z}}\gamma(t,\mathbf{X}(t-),z)\widetilde{\pi}(\mathrm{d}t,\mathrm{d} z),\; t\in[0,T],\\ \mathbf{X}(0)&=\boldsymbol{x} \in\mathbb{H},\end{aligned} \right.\end{equation*} with fully locally monotonecoefficients in a Gelfand triplet $\mathbb{V}\subset\mathbb{H}\subset\mathbb{V}^*$, where the mappings \begin{align*}\mathrm{A}:[0,T]\times \mathbb{V}\to\mathbb{V}^*,\quad \mathrm{B}:[0,T]\times\mathbb{V}\to\mathrm{L}_2(\mathbb{U},\mathbb{H}), \quad\gamma:[0,T]\times\mathbb{V}\times\mathrm{Z}\to\mathbb{H}, \end{align*} aremeasurable, $\mathrm{L}_2(\mathbb{U},\mathbb{H})$ is the space of allHilbert-Schmidt operators from $\mathbb{U}\to\mathbb{H}$, $\mathrm{W}$ is a$\mathbb{U}$-cylindrical Wiener process and $\widetilde{\pi}$ is a compensatedtime homogeneous Poisson random measure. Such kind of SPDE cover a large classof quasilinear SPDE and a good number of fluid dynamic models. Under certaingeneric assumptions of $\mathrm{A},\mathrm{B}$ and $\gamma$, using theclassical Faedo-Galekin technique, a compactness method and a version ofSkorokhod's representation theorem, we prove the existence of a\emph{probabilistic weak solution} as well as \emph{pathwise uniqueness ofsolution}. We use the classical Yamada-Watanabe theorem to obtain the existenceof a \emph{unique probabilistic strong solution}. Finally, we allow bothdiffusion coefficient $\mathrm{B}(t,\cdot)$ and jump noise coefficient$\gamma(t,\cdot,z)$ to depend on both $\mathbb{H}$-norm and $\mathbb{V}$-norm,which implies that both the coefficients could also depend on the gradient ofsolution. We establish the global solvability results.

Well-posedness of a class of stochastic partial differential equations with fully monotone coefficients perturbed by L\'evy noise