$L$-functions for $\mathrm{S}\mathrm{p}(2n)\times \mathrm{G}\mathrm{L}(k)$ via non-unique models

Let $n$ and $k$ be positive integers such that $n$ is even. We derive new global integrals for ${\mathrm{S}\mathrm{p}}_{2n}\times {\mathrm{G}\mathrm{L}}_k$ from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan, following a strategy and extending a previous result of Ginzburg and Soudry on the case $n=k=2$. We show that these new integrals unfold to non-unique models on ${\mathrm{S}\mathrm{p}}_{2n}$. Using the New Way method of Piatetski-Shapiro and Rallis, we show that these new global integrals represent the $L$-functions for ${\mathrm{S}\mathrm{p}}_{2n}\times {\mathrm{G}\mathrm{L}}_k$, generalizing a previous result of the second-named author on ${\mathrm{S}\mathrm{p}}_4\times {\mathrm{G}\mathrm{L}}_2$ and a previous work of Piatetski-Shapiro and Rallis on ${\mathrm{S}\mathrm{p}}_{2n}\times {\mathrm{G}\mathrm{L}}_1$.