Interesting Harmonic Sum $\sum_{k\geq 1}\frac{(-1)^{k-1}}{k^2}H_k^{(2)}$

Question

Here http://integralsandseries.prophpbb.com/topic119.html

We came across the following harmonic sum

$$\tag{1} \sum_{k\geq 1}\frac{(-1)^{k-1}}{k^2}H_k^{(2)}$$

Note that we define

$$H_k^{(2)}=\sum_{n\geq 1}^k\frac{1}{n^2} $$

Also we have

$$\psi_1(k+1)= \zeta(2) -H_k^{(2)} $$

Any ideas how to evaluate (1) ?

Answer 1

A related problem. You can have the following identity

$$\sum_{k=1}^{\infty}(-1)^{k-1} \frac{H_k^{(2)}}{k^2} = \frac{37}{16}\zeta(4)+2\sum_{k=1}^{\infty}(-1)^k \frac{H_k}{k^3}\sim 0.7843781621 .$$