Series with power of generalized harmonic number $\displaystyle\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k$

Question

It's possible to generalize these series? $$\sum_{k=1}^{\infty}H_k^{(s)}x^k=\frac{\operatorname{Li}_s(x)}{1-x}$$ $$\sum_{k=1}^{\infty}H_k^2 x^k=\frac{\ln(1-x)^2+\operatorname{Li}_2(x)}{1-x}$$ Where: $$H_k:=\sum_{j=1}^{k}\frac{1}{j}\text{ are the harmonic numbers}$$ $$H_k^{(s)}:=\sum_{j=1}^{k}\frac{1}{j^s}\text{ are the generalized harmonic numbers}$$ $$\operatorname{Li}_s(z)\text{ is the polylogarithm}$$ I'd like to know if there are formulas for series like this: $$\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k\qquad \text{where }n\in\mathbb{N}$$