  Searches for electroweak production of chargino pairs,
$\tilde{\chi}^{+}_{1}\tilde{\chi}^{-}_{1}$, and of chargino and
next-to-lightest neutralino, $\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{2}$, are
presented. The models explored assume that the charginos decay into a $W$ boson
and the lightest neutralino, $\tilde{\chi}^{\pm}_1 \rightarrow W^{\pm}
\tilde{\chi}^{0}_{1}$. The next-to-lightest neutralinos are degenerate in mass
with the chargino and decay to $\tilde{\chi}^{0}_{1}$ and either a $Z$ or a
Higgs boson, $\tilde{\chi}^{0}_{2} \rightarrow Z \tilde{\chi}^{0}_{1}$ or $h
\tilde{\chi}^{0}_{1}$. The searches exploit the presence of a single isolated
lepton and missing transverse momentum from the $W$ boson decay products and
the lightest neutralinos, and the presence of jets from hadronically decaying
$Z$ or $W$ bosons or from the Higgs boson decaying into a pair of $b$-quarks.
The searches use 139 fb$^{-1}$ of $\sqrt{s}= 13$ TeV proton-proton collisions
data collected by the ATLAS detector at the Large Hadron Collider between 2015
and 2018. No deviations from the Standard Model expectations are found, and 95%
confidence level exclusion limits are set. Chargino masses ranging from 260 to
520 GeV are excluded for a massless $\tilde{\chi}^{0}_{1}$ in chargino pair
production models. Degenerate chargino and next-to-lightest neutralino masses
ranging from 260 to 420 GeV are excluded for a massless $\tilde{\chi}^{0}_{1}$
for $\tilde{\chi}^{0}_{2} \rightarrow Z \tilde{\chi}^{0}_{1}$. For decays
through an on-shell Higgs boson and for mass-splitting between
$\tilde{\chi}^{\pm}_{1} / \tilde{\chi}^{0}_{2}$ and $\tilde{\chi}^{0}_{1}$ as
small as the Higgs boson mass, mass limits are improved by up to 40 GeV in the
range of 200-260 GeV and 280-470 GeV compared to previous ATLAS constraints.
