  A search for a new heavy scalar particle $X$ decaying into a Standard Model
(SM) Higgs boson and a new singlet scalar particle $S$ is presented. The search
uses a proton-proton ($pp$) collision data sample with an integrated luminosity
of 140 fb$^{-1}$ recorded at a centre-of-mass energy of $\sqrt{s} = 13$ TeV
with the ATLAS detector at the Large Hadron Collider. The most sensitive mass
parameter space is explored in $X$ mass ranging from 500 to 1500 GeV, with the
corresponding $S$ mass in the range 200-500 GeV. The search selects events with
two hadronically decaying $\tau$-lepton candidates from $H\to \tau^+\tau^-$
decays and one or two light leptons ($\ell=e,\,\mu$) from $S\to VV$ ($V =
W,\,Z$) decays while the remaining $V$ boson decays hadronically or to
neutrinos. A multivariate discriminant based on event kinematics is used to
separate the signal from the background. No excess is observed beyond the
expected SM background and 95% confidence level upper limits between 72 fb and
542 fb are derived on the cross-section $\sigma(pp\to X\to SH)$ assuming the
same SM-Higgs boson-like decay branching ratios for the $S\to VV$ decay. Upper
limits on the visible cross-sections $\sigma(pp\to X\to SH \to WW\tau\tau)$ and
$\sigma(pp\to X\to SH \to ZZ\tau\tau)$ are also set in the ranges 3-26 fb and
6-33 fb, respectively.
