  Hidden-charm exotic hadrons will be searched for and investigated at future
electron-ion colliders. For instance, the $X(3872)$ can be produced through the
exclusive process $\gamma p\to X(3872)p$. The vector meson dominance model has
been commonly employed in estimating the cross sections of such processes.
However, the coupled-channel production mechanism through open-charm
meson-baryon intermediate states may play a crucial role. To assess the
significance of such contributions, we estimate the cross section of the
$\gamma p\to X(3872)p$ reaction assuming the coupled-channel mechanism. For
energies near the threshold, the total cross section is predicted to be of tens
of nanobarns for $\gamma p\to X(3872)p$, which can be measured at future
experimental facilities. Furthermore, the open-charm coupled-channel mechanism
leads to a distinct line shape of the total cross section that can be utilized
to reveal the production dynamics.
