The Yamada-Watanabe theory provides a robust framework for understanding stochastic equations driven by Wiener processes. Despite its comprehensive treatment in the literature, the applicability of the theory to SPDEs driven by Poisson random measures or, more generally, Lévy processes remains significantly less explored, with only a handful of results addressing this context. In this work, we leverage a result by Kurtz to demonstrate that the existence of a martingale solution combined with pathwise uniqueness implies the existence of a unique strong solution for SPDEs driven by both a Wiener process and a Poisson random measure. Our discussion is set within the variational framework, where the SPDE under consideration may be nonlinear. This work is influenced by earlier research conducted by the second author alongside de Bouard and Ondreját.