The production of Dark Matter in association with Higgs bosons is predicted in several extensions of the Standard Model. An exploration for such scenarios is presented, considering final states with missing transverse momentum and $b$-tagged jets consistent with a Higgs boson. The analysis uses proton-proton collision data at a centre-of-mass energy of 13 TeV recorded by the ATLAS experiment at the LHC during Run 2, amounting to an integrated luminosity of 139 fb$^{-1}$. Besides the increased dataset, the analysis also presents further improvements with respect to previous searches, providing sensitivity to a wider class of possible signal scenarios. These improvements include both an optimised event selection and advances in the object identification, such as the use of the likelihood-based significance of the missing transverse momentum and variable-radius track jets. No significant deviation from Standard Model expectations is observed. The results are interpreted in two benchmark models with two Higgs doublets extended by either a heavy vector boson $Z'$ or a pseudoscalar singlet $a$ and which provide a dark matter candidate $\chi$. In the case of the Two-Higgs-Doublet model with an additional vector boson $Z'$, the observed limits extend up to a $Z'$ mass of 3.1 TeV at 95 \% confidence level for a mass of $100$ GeV for the Dark Matter candidate. For the Two-Higgs-Doublet model with an additional pseudoscalar $a$, masses of $a$ are excluded up to 520 GeV and 240 GeV for $\tan \beta = 1$ and $\tan \beta = 10$ and a Dark Matter mass of $10$ GeV, respectively. Limits on the visible cross sections are set and range from to 0.05 fb to 3.26 fb, depending on the missing transverse momentum and $b$-quark jet multiplicity requirements.

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