  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 $11^\circ$
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 $m_G\approx 1.4$ 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
$Br(B\to\eta'K)$. We then analyze the $\eta_c$-mixing effects on charmonium
magnetic dipole transitions, and on the $B\to\eta^{(\prime)}K_S$ 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 $Br(B\to\eta'K)$
by 18%, but does not alter those for $Br(B\to\eta K)$. The puzzle due to the
large $Br(B\to\eta'K)$ data is then resolved.
