  A search for the exotic decay of the Higgs boson ($H$) into a $b\bar{b}$
resonance plus missing transverse momentum is described. The search is
performed with the ATLAS detector at the Large Hadron Collider using 139
$\mathrm{fb}^{-1}$ of $pp$ collisions at $\sqrt{s} = 13$ TeV. The search
targets events from $ZH$ production in an NMSSM scenario where $H \rightarrow
\tilde{\chi}^{0}_{2}\tilde{\chi}^{0}_{1}$, with $\tilde{\chi}^{0}_{2}
\rightarrow {a} \tilde{\chi}^{0}_{1}$, where $a$ is a light pseudoscalar Higgs
boson and $\tilde{\chi}^{0}_{1,2}$ are the two lightest neutralinos. The decay
of the $a$ boson into a pair of $b$-quarks results in a peak in the dijet
invariant mass distribution. The final-state signature consists of two leptons,
two or more jets, at least one of which is identified as originating from a
$b$-quark, and missing transverse momentum. Observations are consistent with
Standard Model expectations and upper limits are set on the product of cross
section times branching ratio for a three-dimensional scan of the masses of the
$\tilde{\chi}^{0}_{2}$, $\tilde{\chi}^{0}_{1}$ and $a$ boson.
