A multivariate integral representation on $\mathrm{GL}_2 \times \mathrm{GSp}_4$ inspired by the pullback formula

We give a two variable Rankin-Selberg integral inspired by consideration of Garrett's pullback formula. For a globally generic cusp form on $\mathrm{GL}_2\times \mathrm{GSp}_4$, the integral represents the product of the $\mathrm{Std}\times \mathrm{Spin}$ and $\mathbf{1} \times \mathrm{Std}$ $L$-functions. We prove a result concerning an Archimedean principal series representation in order to verify a case of Jiang's first-term identity relating certain non-Siegel Eisenstein series on symplectic groups. Using it, we obtain a new proof of a known result concerning possible poles of these $L$-functions.