Evaluations of ${\sum }_{k=1}^{\mathrm{\infty }}\frac{x^k}{k^2\left(\genfrac{}{}{0px}{}{3k}{k}\right)}$ and related series

We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun’s related works in 2022–2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum _{k=0}^{\mathrm{\infty }}\frac{x_0^k}{(k+1)\left(\genfrac{}{}{0px}{}{3k}{k}\right)}\text{ and }\sum _{k=1}^{\mathrm{\infty }}\frac{x_0^k}{k^2\left(\genfrac{}{}{0px}{}{3k}{k}\right)}$$ for any $x_0\in (-27\mathrm{/}4,27\mathrm{/}4)$.