$L$-functions for $\mathrm{Sp}(2n)\times\mathrm{GL}(k)$ via non-unique models

Let $n$ and $k$ be positive integers such that $n$ is even. We derive new global integrals for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ 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 $\mathrm{Sp}_{2n}$. Using the New Way method of Piatetski-Shapiro and Rallis, we show that these new global integrals represent the $L$-functions for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$, generalizing a previous result of the second-named author on $\mathrm{Sp}_{4}\times\mathrm{GL}_2$ and a previous work of Piatetski-Shapiro and Rallis on $\mathrm{Sp}_{2n}\times\mathrm{GL}_1$.