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

A related problem. You can have the following identity

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