A search for dark matter produced in association with a Higgs boson decaying to $b$-quarks is performed using $pp$ collisions at a centre-of-mass energy of \sqrt{s}= 13\,TeV. The dataset has an integrated luminosity of 79.8\,fb$^{-1}$ and was recorded with the ATLAS detector at the Large Hadron Collider. Selected collision events comprise large missing transverse momentum and either two $b$-tagged small radius jets or a single large radius jet containing two $b$-tagged subjets. The reconstruction of these subjets is based on a jet algorithm that uses inner detector tracks as inputs and a radius parameter that decreases as the subjet transverse momentum increases. The observed data are in agreement with the Standard Model predictions. The results are interpreted in the context of a simplified model ($Z^{\prime}$-2HDM) which describes the interaction of dark matter and Standard Model particles via new heavy mediator particles. Masses of the $Z^{\prime}$ are excluded up to 2.8\,TeV depending on the choices for other model parameters.

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