Mixed moments of $\mathrm{G}\mathrm{L}(2)$ and symmetric square $L$-functions

In this paper, we prove asymptotic formulas of mixed moments of $\mathrm{G}\mathrm{L}(2)$ and its symmetric square $L$-functions for both Hecke–Maass cusp forms and holomorphic Hecke eigenforms in short intervals. As an application, we prove quantitative simultaneous non-vanishing of central values of these $L$-functions.