  This article presents a search for new resonances decaying into a $Z$ or $W$
boson and a 125 GeV Higgs boson $h$, and it targets the $\nu\bar{\nu}b\bar{b}$,
$\ell^+\ell^-b\bar{b}$, or $\ell^{\pm}{\nu}b\bar{b}$ final states, where
$\ell=e$ or $\mu$, in proton-proton collisions at $\sqrt{s}=13$ TeV. The data
used correspond to a total integrated luminosity of 139 fb$^{-1}$ collected by
the ATLAS detector during Run 2 of the LHC at CERN. The search is conducted by
examining the reconstructed invariant or transverse mass distributions of $Zh$
or $Wh$ candidates for evidence of a localised excess in the mass range from
220 GeV to 5 TeV. No significant excess is observed and 95% confidence-level
upper limits between 1.3 pb and 0.3 fb are placed on the production cross
section times branching fraction of neutral and charged spin-1 resonances and
CP-odd scalar bosons. These limits are converted into constraints on the
parameter space of the Heavy Vector Triplet model and the two-Higgs-doublet
model.
