Search for $1^{-+}$ charmonium-like hybrid via $e^{+}e^{-}\rightarrow γη^{(\prime)} η_{c}$ at center-of-mass energies between 4.258 and 4.681 GeV

Using $e^{+}e^{-}$ collision data corresponding to an integrated luminosity of 10.6 fb$^{-1}$ collected at center-of-mass energies between 4.258 and 4.681 GeV with the BESIII detector at the BEPCII collider, we search for the $1^{- +}$ charmonium-like hybrid via $e^{+}e^{-}\rightarrowγηη_{c}$ and $e^{+}e^{-}\rightarrowγη^{\prime}η_{c}$ decays for the first time. No significant signal is observed and the upper limits on the Born cross sections for both processes are set at the 90% confidence level.