A search for the resonant production of a heavy scalar X decaying into a Higgs boson and a new lighter scalar S, through the process X → S(→$ b\overline{b} $)H(→γγ), where the two photons are consistent with the Higgs boson decay, is performed. The search is conducted using an integrated luminosity of 140 fb$^{−1}$ of proton-proton collision data at a centre-of-mass energy of 13 TeV recorded with the ATLAS detector at the Large Hadron Collider. The search is performed over the mass range 170 ≤ m$_{X}$ ≤ 1000 GeV and 15 ≤ m$_{S}$ ≤ 500 GeV. Parameterised neural networks are used to enhance the signal purity and to achieve continuous sensitivity in a domain of the (m$_{X}$, m$_{S}$) plane. No significant excess above the expected background is found and 95% CL upper limits are set on the cross section times branching ratio, ranging from 39 fb to 0.09 fb. The largest deviation from the background-only expectation occurs for (m$_{X}$, m$_{S}$) = (575, 200) GeV with a local (global) significance of 3.5 (2.0) standard deviations.