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 $pi^0(eta)rightarrowgammagamma$ 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.
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