This note presents an investigation of the $CP$ properties of the coupling between the Higgs boson and the top quark, employing 139 fb$^{-1}$ of proton--proton collision data recorded by the ATLAS experiment at a centre-of-mass energy of $\sqrt{s}=13$ TeV. The $CP$ structure of the top-Higgs boson Yukawa coupling is probed in events with a Higgs boson decaying to a pair of $b$ quarks and produced in association with a pair of top quarks, $t\bar{t}H$, or a single top quark, $tH$. Events containing one or two electrons or muons are used for the measurement. In an extension of the Standard Model with a CP-odd admixture to the top-Higgs Yukawa coupling, the mixing angle between $CP$-even and $CP$-odd couplings is measured to be \alpha=11^{\circ+55^{\circ}}_{~-77^{\circ}}. A pure $CP$-odd coupling is disfavoured by the data at 1.2$\sigma$ confidence level.

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