Infinite summation giving wierd results

We are searching in our group for closed forms of derivatives of hypergeometric functions. This leads to expressions like

$ \sum\limits_{m=2}^\infty \frac{z^m\Gamma[m-1/2]H_m}{2m^2\sqrt{\pi}\Gamma[m]}$

where $ H_m$ denotes the m-th harmonic number. Now trying to evaluate this in Mathematica 12.0 ( or 12.1.1) using

Sum[(z^m*Gamma[-(1/2) + m]*HarmonicNumber[m])/(2*m^2*Sqrt[Pi]*Gamma[m]), {m, 2, Infinity}]

returns 0. But in this case we actually know a rather complicated closed form expression for this sum in terms of logs and polylogs which are non-vanishing. Moreover, taking the case z=1, Mathematica 12.0 evaluates the sum correctly, i.e.

Sum[Gamma[-(1/2) + m]*HarmonicNumber[m])/(2*m^2*Sqrt[Pi]*Gamma[m]), {m, 2, Infinity}]

returns $ \frac{7 \sqrt{\pi }-\frac{2 \pi ^{5/2}}{3}}{2 \sqrt{\pi }}$ which is correct and non-zero. Thus the result form the original command seams to be wrong. Are we missing something? Is there a way to prevent these wrong evaluations? We would like to use Mathematica to compute some series with a priori unknown closed forms and that behaviour is worrying us.