$p$-adic $L$-functions for $\mathrm{G}\mathrm{S}\mathrm{p}(4)\times \mathrm{G}\mathrm{L}(2)$

We construct $p$-adic $L$-functions interpolating the critical values of the degree eight $L$-functions of $\mathrm{G}\mathrm{S}\mathrm{p}(4)\times \mathrm{G}\mathrm{L}(2)$ for cuspidal automorphic representations generated by $p$-ordinary Siegel modular forms of genus two and $p$-ordinary modular forms.