  Production of $\pi^0$ and $\eta$ mesons has been measured at midrapidity in
Cu$+$Au collisions at $\sqrt{s_{_{NN}}}$=200 GeV. Measurements were performed
in $\pi^0(\eta)\rightarrow\gamma\gamma$ decay channel in the 1(2)-20 GeV/$c$
transverse momentum range. A strong suppression is observed for $\pi^0$ and
$\eta$ meson production at high transverse momentum in central Cu$+$Au
collisions relative to the $p$$+$$p$ results scaled by the number of
nucleon-nucleon collisions. In central collisions the suppression is similar to
Au$+$Au with comparable nuclear overlap. The $\eta/\pi^0$ ratio measured as a
function of transverse momentum is consistent with $m_T$-scaling
parameterization down to $p_T=$2 GeV/$c$, its asymptotic value is constant and
consistent with Au$+$Au and $p$$+$$p$ and does not show any significant
dependence on collision centrality. Similar results were obtained in
hadron-hadron, hadron-nucleus, and nucleus-nucleus collisions as well as in
$e^+e^-$ collisions in a range of collision energies $\sqrt{s_{_{NN}}}=$3--1800
GeV. This suggests that the quark-gluon-plasma medium produced in Cu$+$Cu
collisions either does not affect the jet fragmentation into light mesons or it
affects the $\pi^0$ and $\eta$ the same way.
