  We derive new integrals for $\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ from the
generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan for any
positive even integer $n$ and any positive integer $k$, following a strategy
and extending a previous result of Ginzburg and Soudry. We show that these new
integrals unfold to non-unique models on $\mathrm{Sp}_{2n}$. When $k$ divides
$n$, we carry out the unramified computation and show that these integrals
represent the tensor product $L$-function for
$\mathrm{Sp}_{2n}\times\mathrm{GL}_k$ via the New Way method, generalizing a
previous result of the second-named author on
$\mathrm{Sp}_{4}\times\mathrm{GL}_2$.
