We study the scalar flavor-singlet meson spectrum using a basis of operators which includes light-meson, charmonium, two-pion and purely gluonic ones. We use improved distillation to build meson operators which create states that better resemble the energy eigenstates of interest. This is done in $N_f = 3 + 1$ QCD with a close-to-physical charm quark and three degenerate light quarks at $m_{\pi}= 420, 800$ MeV.  We observe non-zero mixing correlations between the different types of operators, particularly between light meson and charmonium ones when taking into account charm annihilation effects, and use a GEVP formulation to quantify the overlaps between the states created by our operators and the energy eigenstates of the theory. We include two-pion operators in our basis because we are particularly interested in the scalar glueball and its possible decay into pions.