  The $\eta_c$ photoproduction in $ep$ collision has long been proposed as an
ideal process to probe the existence of odderon. In the current work, we
systematically investigate the photoproduction of various $C$-even heavy
quarkonia (exemplified by $\eta_{c(b)}$, and $\chi_{c(b)J}$ with $J=0,1,2$) via
one-photon exchange channel, at the lowest order in $\alpha_s$ and heavy quark
velocity in the context of NRQCD factorization. We find that the
photoproduction rates of the $C$-even quarkonia through this mechanism are
comparable in magnitude with that through the odderon-initiated mechanism, even
in the Regge limit ($s\gg -t$), though the latter types of predictions suffers
from considerable theoretical uncertainties. The future measurements of these
types of quarkonium photoproduction processes in \texttt{EIC} and \texttt{EicC}
are crucial to ascertain which mechanism plays the dominant role.
