A search for exotic Higgs boson decays $\mathrm{H} \to \mathcal{A} \mathcal{A}$, with $\mathcal{A} \to \gamma \gamma$ is presented, using events with a semi-merged topology. One of the hypothetical particles, $\mathcal{A}$, is assumed to decay promptly into a semi-merged diphoton system reconstructed as a single photon-like object, while the other $\mathcal{A}$ decays into two resolved photons. The search is performed using proton-proton collision data collected by the CMS experiment at $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of $138~fb^{-1}$. The data agree with the standard model background expectation.Upper limits are set on the product of the Higgs boson production cross section and the branching fraction, $\sigma(\mathrm{pp} \to \mathrm{H})\,\mathcal{B}( \mathrm{H} \to \mathcal{A} \mathcal{A} \to 4\gamma)$, which range from 0.264 to 0.005$\,\mathrm{pb}$ at 95% confidence level, for $\mathcal{A}$ masses in the range $1 < m_\mathcal{A} < 15$ GeV. These limits are the most stringent to date in the 1--5 GeV $m_\mathcal{A}$ range.