  We present an improved analysis of our lattice data for the $\eta$--$\eta'$
system, including a correction of the relevant correlation functions for
residual topological finite size effects and employing consistent chiral and
continuum fits. From this analysis we update our physical results for the
masses $M_\eta=557(11)_\mathrm{stat}(03)_{\chi\mathrm{PT}}\,\mathrm{MeV}$ and
$M_{\eta'}=911(64)_\mathrm{stat}(03)_{\chi\mathrm{PT}}\,\mathrm{MeV}$, as well
as the mixing angle in the quark flavor basis
$\phi=38.8(2.2)_\mathrm{stat}(2.4)_{\chi\mathrm{PT}}^\circ$ in excellent
agreement with other results from phenomenology. Similarly, we include an
analysis for the decay constant parameters, leading to
$f_l=125(5)_\mathrm{stat}(6)_{\chi\mathrm{PT}}\,\mathrm{MeV}$ and
$f_s=178(4)_\mathrm{stat}(1)_{\chi\mathrm{PT}}\,\mathrm{MeV}$. The second error
reflects the uncertainty related to the chiral extrapolation. The data used for
this study has been generated on gauge ensembles provided by the European
Twisted Mass Collaboration with $N_f=2+1+1$ dynamical flavors of Wilson twisted
mass fermions. These ensembles cover a range of pion masses from
$220\,\mathrm{MeV}$ to $500\,\mathrm{MeV}$ and three values of the lattice
spacing. Combining our data with a prediction from chiral perturbation theory,
we give an estimate for the physical $\eta,\eta' \rightarrow \gamma\gamma$
decay widths and the singly-virtual $\eta,\eta'\rightarrow\gamma\gamma^*$
transition form factors in the limit of large momentum transfer.
