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For $f$ a holomorphic cusp form of weight $k$ and full level and $g$ a primitive holomorphic cusp form of weight $2 \\kappa$ and level $P$, we prove hybrid subconvexity bounds for $L \\left(\\tfrac{1}{2}, \\text{Sym}^2 f \\otimes g\\right)$ in the $k$ and $P$ aspects when $P^{\\frac {13} {64} + \\delta} < k < P^{\\frac 3 8 - \\delta}$ for any $0 < \\delta < \\frac {11} {128}$. 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