Are there any general formula for the following series

$$\tag{1}\sum_{n\geq 1}\frac{H^{(p)}_nH_n}{n^q}$$

Where we define

$$H^{(p)}_n= \sum_{k=1}^n \frac{1}{k^p}\,\,\,\,\,H^{(1)}_n\equiv H_n =\sum_{k=1}^n\frac{1}{k} $$

For the special case $p=q=2$ in (1) I found the following paper

Stating that

$$\sum_{n\geq 1}\frac{H^{(2)}_nH_n}{n^2}=\zeta(5)+\zeta(2)\zeta(3)$$

See equation (3a) .

Is there any other paper in the literature discussing (1) or any special cases ?