We study $e^{+}e^{-}$ collisions with a $pi^{+}pi^{-}pi^{0}eta_{c}$ final state using data samples collected with the BESIII detector at center-of-mass energies $sqrt{s}=4.226$, $4.258$, $4.358$, $4.416$, and $4.600$ GeV. Evidence for the decay $zcpmtorhopmetac$ is reported with a statistical significance of $3.9sigma$ with various systematic uncertainties taken into account at $sqrt{s} = 4.226$ GeV, and the Born cross section times branching fraction $sigma^{B}(EEto pimpzcpm)times BR(zcpmtorhopmetac)$ is measured to be $(48 pm 11 pm 11),rm{pb}$. The $zcpmto rhopmetac$ signal is not significant at the other center-of-mass energies and the corresponding upper limits are determined. In addition, no significant signal is observed in a search for $zcppmto rho^{pm}etac$ with the same data samples. The ratios $R_{zc}=BR(zcpmto rho^{pm} etac)/BR(zcpmto pi^{pm} jpsi)$ and $R_{zcp}=BR(zcppmto rho^{pm} etac)/BR(zcppmto pi^{pm} hc)$ are obtained and used to discriminate between different theoretical interpretations of the $zcpm$ and $zcppm$.
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