By using a 2.92 fb$^{-1}$ data sample taken at $sqrt{s} = 3.773$ GeV with the BESIII detector operating at the BEPCII collider, we search for the radiative transitions $psi(3770)togammaeta_c$ and $gammaeta_c(2S)$ through the hadronic decays $eta_c(et
a_c(2S))to K^0_SK^pmpi^mp$. No significant excess of signal events above background is observed. We set upper limits at a 90% confidence level for the product branching fractions to be $mathcal{B}(psi(3770)togammaeta_c)times mathcal{B}(eta_cto K^0_SK^pmpi^mp) < 1.6times10^{-5}$ and $mathcal{B}(psi(3770)togammaeta_c(2S))times mathcal{B}(eta_c(2S)to K^0_SK^pmpi^mp) < 5.6times10^{-6}$. Combining our result with world-average values of $mathcal{B}(eta_c(eta_c(2S))to K^0_SK^pmpi^mp)$, we find the branching fractions $mathcal{B}(psi(3770)togammaeta_c) < 6.8times10^{-4}$ and $mathcal{B}(psi(3770)togammaeta_c(2S)) < 2.0times10^{-3}$ at a 90% confidence level.
