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In this paper we prove a conjecture of Ginzburg and Soudry on an integralrepresentation for the $L$-function $L^S(s, \pi\times \tau)$ attached to a pair$(\pi, \tau)$ of irreducible automorphic cuspidal representations of$\mathrm{Sp}_4({\mathbb A})$ and $\mathrm{GL}_2({\mathbb A})$, which is derivedfrom the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Weshow that the integral unfolds to a non-unique model and analyze it using theNew Way method of Piatetski-Shapiro and Rallis. As applications, we determinethe poles of $L^S(s, \pi\times \tau)$ and relate the existence of the poles tothe non-vanishing of certain period integrals. Moreover, for certain family ofcuspidal representations $\tau$ of $\mathrm{GL}_2({\mathbb A})$, we prove that$L^S(s, \pi\times \tau)$ is holomorphic.

$L$-function for $\mathrm{Sp}(4)\times\mathrm{GL}(2)$ via a non-unique model