A note on some infinite sums of Hurwitz zeta functions

We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $\zeta(s,\alpha)$ of the form $$\sum_{k=1}^\infty (\pm 1)^k k^m \zeta(s,k),$$ where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $\zeta(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.