Context. The interplay between star formation and supermassive-black-hole growth is central to galaxy evolution. While X-ray-selected active glactic nuclei AGNs preferentially reside in star-forming galaxies, the role of host-galaxy morphology in regulating both star-formation enhancement and AGN triggering across the star-forming main sequence remains unclear. Aims. We aim to investigate the star formation properties and incidence of X-ray AGNs across the star-forming main sequence using the DESI─eRASS1 dataset, focusing on the role of host-galaxy structure. Methods. Our analysis includes 1171 X-ray-selected AGNs and 45 374 non-AGN star-forming galaxies at z ≤ 1.5. We quantified star formation in AGN hosts relatively to matched control samples using the SFRnorm parameter and examined its dependence on the X-ray luminosity (LX) and specific accretion rate (λsBHAR). We also measured AGN incidence as a function of distance from the star-forming main sequence (∆MS), separating disk- and spheroid-dominated systems based on Sérsic-index classifications. Results. The SFRnorm remains close to unity at low to intermediate LX and increases at higher luminosities, with the transition shifting towards higher LX in more massive systems. This trend depends on morphology: in the stellar-mass range of 10.5 ≤ log(M★/M⊙) < 11.5, disk-dominated AGN hosts exhibit enhanced SFRnorm at moderate LX, while spheroid-dominated systems remain consistent with unity. No comparable morphology dependence is found when using λsBHAR. The incidence of X-ray AGNs increases strongly with ∆MS in both redshift bins, with a steeper dependence at higher redshift. The ∆MS─incidence relation is also morphology dependent and evolves with redshift. Conclusions. The connection between star formation and AGN activity is primarily governed by global gas availability but modulated by host-galaxy structure and cosmic epoch. Absolute AGN power is more tightly linked to host-wide star formation than accretion efficiency normalised by stellar mass.