  We investigate the semi-inclusive production of hidden-charm exotic states,
including $X(3872)$, $Z_c$, $Z_{cs}$ and the pentaquark $P_c$ states, in
lepton-proton scattering processes. These hadrons are close to the thresholds
of a pair of charm and anticharm hadrons, and are assumed to have hadronic
molecular structure as their main components. In order to give
order-of-magnitude estimates of the cross sections, we use Pythia to simulate
the short-distance productions of the constituent hadrons, which then rescatter
to form the exotic hadrons. The estimates for the $X(3872)$ and $Z_c(3900)^\pm$
are not in conflict with the upper limits measured at the COMPASS experiment
for the exclusive photoproduction process. The results here indicate that the
considered hidden-charm states can be copiously produced at the proposed
electron-ion colliders EicC and US-EIC.
