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

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