A study of the structure of the coupling between the Higgs boson and the top quark is performed using events from $t\bar{t}H$ and $tH$ production in the $H\to \gamma \gamma$ decay channel, with 164 fb${}^{-1}$ of proton-proton collision data at a center-of-mass energy of $\sqrt{s}$ = 13.6 TeV collected by the ATLAS detector at the LHC. The cross section of the $t\bar{t}H$ process times the Higgs to diphoton decay branching ratio is measured to be ${1.46}_{-0.35}^{+0.40}={1.46}_{-0.32}^{+0.34}{\mathrm{\text{(stat.)}}}_{-0.13}^{+0.22}\mathrm{\text{(sys.)}}$ fb, corresponding to ${1.13}_{-0.28}^{+0.33}$ times the Standard Model prediction. An observed 95% confidence level limit on the $tH$ production cross section times the Higgs to diphoton decay branching ratio is set at 6.2 times the Standard Model prediction, compared to an expected limit of 4.4 times, constituting the most stringent $tH$ upper limit achieved in a single measurement to date. The results are combined with 140 fb${}^{-1}$ of proton-proton collision data collected at $\sqrt{s}$ = 13 TeV in the same production and decay channel, and a $CP$-mixing angle of $|\alpha |>{38}^{\circ }$ is excluded at the 95% confidence level, with a purely $CP$-odd Higgs-top Yukawa coupling excluded at the level of 5.8 standard deviations, providing the most stringent direct constraints on the $CP$ structure of the Higgs–top Yukawa interaction to date.