How to automate calculations of sums with harmonic numbers

I would like to ask if anyone knows how can we automate calculations of the following sum

Sum[HarmonicNumber[n]^2/((n + 1) 2^n), {n, 1, ∞}]

The analytic result is known and can be verified numerically

NSum[HarmonicNumber[n]^2/((n + 1) 2^n), {n, 1, ∞}]

0.650161

Log[2]^3/3 + Zeta[2] Log[2] - Zeta[3]/2 // N

0.650161

I would be most interested to see generalizations to other powers $k$

$$sum_{n=1}^{infty}frac{H_{n}^k}{(n+1)2^n} $$

It might be tempting to say that this question belongs to other pure math-orient stackexchange site. However, analytic calculations are so cumbersome that some help of a computer algebra system is needed.

Mathematical background:

These sums have applications in the calculations of Feynman integrals. Some starting point could be this integral representation of the harmonic number (Euler):
$$ H_n = int_0^1 frac{1 – x^n}{1 – x},dx. $$