Metallicity offers a unique window into the baryonic history of the cosmos, being instrumental in probing evolutionary processes in galaxies between different cosmic environments. We aim to quantify the contribution of these environments to the scatter in the mass-metallicity relation (MZR) of galaxies. By analysing the galaxy distribution within the cosmic skeleton of the Horizon Run 5 cosmological hydrodynamical simulation at redshift z = 0.625, computed using a careful calibration of the T-ReX filament finder, we identify galaxies within three main environments: nodes, filaments, and voids. We also classify galaxies based on the dynamical state of the clusters and the length of the filaments in which they reside. We find that the cosmic environment significantly contributes to the scatter in the MZR; in particular, both the gas metallicity and its average relative standard deviation increase when considering denser large-scale environments. The difference in the average metallicity between galaxies within relaxed and unrelaxed clusters is ≈0.1dex, with both populations displaying positive residuals, δZ_g, from the averaged MZR. Moreover, the difference in metallicity between node and void galaxies accounts for $\approx 0.14 \, \text{dex}$ in the scatter of the MZR at stellar mass $M_{\star } \approx 10^{9.35}\, \text{M}_{\odot }$. Finally, both the average [O/Fe] in the gas and the galaxy gas fraction decrease when moving to higher large-scale densities in the simulation, suggesting that the cores of cosmic environments host - on average - older and more massive galaxies, whose enrichment is affected by a larger number of Type Ia Supernova events.