We investigate the mixing between S-wave flavor-singlet light meson and charmonium operators in two ensembles at different pion masses ($m_{\pi} \approx 420, 800$ MeV). By solving a GEVP we find both types of operators have non-zero overlaps with all states we look at. We also compare the resulting spectrum with the one coming from separate GEVPs including either only light meson or only charmonium operators and quantify the effects of the mixing on the hyperfine splitting. The largest effect we observe is a decrease of the $\eta_c$ mass by 39(24) MeV. Finally, we show preliminary results of the mixing between both types of operators with glueball ones as a first step to further extend our operator basis.