  We study $\eta_{\rm c}$ production at center-of-mass energies $\sqrt{s}$ from
4.18 to 4.60 GeV in $e^+e^-$ annihilation data collected with the BESIII
detector operating at the BEPCII storage ring, corresponding to 7.3 fb$^{-1}$
of integrated luminosity. We measure the cross sections of the three different
exclusive reactions $e^+e^-\rightarrow \eta_{\rm c}\pi^+ \pi^-\pi^0$, $e^+e^-
\rightarrow \eta_{\rm c}\pi^+ \pi^-$, and $e^+e^- \rightarrow \eta_{\rm
c}\pi^0\gamma$. We find significant $\eta_{\rm c}$ production in
$e^+e^-\rightarrow \eta_{\rm c}\pi^+ \pi^-\pi^0$ at $\sqrt{s}$ of 4.23 GeV and
4.26 GeV and observe a significant energy-dependent Born cross section that we
measure to be consistent with the production via the intermediate $Y(4260)$
resonance. In addition, we perform a search for a charmonium-like $Z_{\rm c}$
state close to the $D\bar{D}$ threshold that decays to $\eta_{\rm c}\pi$,
involving ground state charmonium, and observe no signal. Corresponding upper
limits on the cross section of $\eta_{\rm c}$ and $Z_{\rm c}$ production are
provided, where the yields are not found to be significant.
