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 $e^+e^- rightarrow eta_{rm c}pi^0gamma$. 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 $Dbar{D}$ 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.
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