Higher moments of the symmetric square $L$-function off the critical line

Let $f$ be the Hecke eigenform for the modular group $S{L}_2(\mathbb{Z})$, and $L(s,{\text{sym}}^{2}f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<\sigma <1$, define $m(\sigma )$ as the supremum of all numbers $m$ such that $${\int }_1^T\mathrm{\mid }L(\sigma +it,{\text{sym}}^{2}f){\mathrm{\mid }}^m\text{d}t{\ll }_f{T}^{1+\epsilon },$$ where $\epsilon >0$ is an arbitrarily small number. In this paper, we established the bound \begin{align*} m(σ)\geq \frac{17}{26-28σ}, $\text{ for }$\frac{5}{8}\leqσ\leq\frac{52}{73}, \end{align*} which improved our previous result.