  Neutral pion and $\eta$ meson production cross sections were measured up to
unprecedentedly high transverse momenta ($p_{\rm T}$) in p-Pb collisions at
$\sqrt{s_{\rm NN}}$ = 8.16 TeV. The mesons were reconstructed via their
two-photon decay channel in the rapidity interval $-1.3< y <0.3$ in the ranges
of $0.4<p_{\rm T}<200$ GeV/$c$ and $1.0<p_{\rm T}<50$ GeV/$c$, respectively.
The respective nuclear modification factor ($R_{\rm pPb}$) is presented for
$p_{\rm T}$ up to of 200 and 30 GeV/$c$, where the former was achieved
extending the $\pi^{0}$ measurement in pp collisions at $\sqrt{s}$ = 8 TeV. The
values of $R_{\rm pPb}$ are below unity for $p_{\rm T}<10$ GeV/$c$, while they
are consistent with unity for $p_{\rm T}>10$ GeV/$c$. The new data provide
constraints for nuclear parton distribution and fragmentation functions over a
broad kinematic range and are compared to model predictions as well as previous
results at $\sqrt{s_{\rm NN}}$ = 5.02 TeV.
