We include the eta(c) meson into the eta-eta'-G mixing formalism constructed in our previous work, where G represents the pseudoscalar gluball. The mixing angles in this tetramixing matrix are constrained by theoretical and experimental implications from relevant hadronic processes. Especially, the angle between eta(c) and G is found to be about 11 degrees from the measured decay widths of the eta(c) meson. The pseudoscalar glueball mass m(G), the pseudoscalar densities m(qq,ss,cc), and the U(1) anomaly matrix elements associated with the mixed states are solved from the anomalous Ward identities. The solution m(G) approximate to 1.4 GeV obtained from the eta-eta'-G mixing is confirmed, while m(qq) grows to above the pion mass, and thus increases perturbative QCD predictions for the branching ratios Br(B -> eta'K). We then analyze the eta(c)-mixing effects on charmonium magnetic dipole transitions, and on the B -> eta K-(')(S) branching ratios and CP asymmetries, which further improve the consistency between theoretical predictions and data. A predominant observation is that the eta(c) mixing enhances the perturbative QCD predictions for Br(B -> eta'K) by 18%, but does not alter those for Br(B -> eta K). The puzzle due to the large Br(B -> eta'K) data is then resolved.