The results of a search for electroweakino pair production $pp\to \chi ~1\pm \chi ~20$ in which the chargino ($\chi ~1\pm$) decays into a $W$ boson and the lightest neutralino ($\chi ~10$), while the heavier neutralino ($\chi ~20$) decays into the Standard Model 125 GeV Higgs boson and a second $\chi ~10$ are presented. The signal selection requires a pair of $b$-tagged jets consistent with those from a Higgs boson decay, and either an electron or a muon from the $W$ boson decay, together with missing transverse momentum from the corresponding neutrino and the stable neutralinos. The analysis is based on data corresponding to 139 $\mathrm{fb}-1$ of $s=13$ TeV $pp$ collisions provided by the Large Hadron Collider and recorded by the ATLAS detector. No statistically significant evidence of an excess of events above the Standard Model expectation is found. Limits are set on the direct production of the electroweakinos in simplified models, assuming pure wino cross-sections. Masses of $\chi ~1\pm /\chi ~20$ up to 740 GeV are excluded at 95% confidence level for a massless $\chi ~10$.