Hybrid Weyl-type bound for $p$-power twisted $\mathrm{G}\mathrm{L}(2)$ $L$-functions

Let $g$ be a fixed holomorphic cusp form of arbitrary level and nebentypus. Let $\chi$ be a primitive character of prime-power modulus $q=p^{\gamma }$. In this paper, we prove the following hybrid Weyl-type subconvexity bound \begin{align*} L (1/2 + it, g \otimes χ) \ll_{g, p, \varepsilon} ( (1+|t|) q )^{1/3+ \varepsilon} \end{align*} for any $\epsilon >0$.