We report on magnetization $M(H)$, dc/ac magnetic susceptibility $chi(T)$, specific heat $C_{mathrm{m}}(T)$ and muon spin relaxation ($mu$SR) measurements of the Kitaev honeycomb iridate Cu$_2$IrO$_2$ with quenched disorder. In spite of the chemical disorders, we find no indication of spin glass down to 260~mK from the $C_{mathrm{m}}(T)$ and $mu$SR data. Furthermore, a persistent spin dynamics observed by the zero-field muon spin relaxation evidences an absence of static magnetism. The remarkable observation is a scaling relation of $chi[H,T]$ and $M[H,T]$ in $H/T$ with the scaling exponent $alpha=0.26-0.28$, expected from bond randomness. However, $C_{mathrm{m}}[H,T]/T$ disobeys the predicted universal scaling law, pointing towards the presence of low-lying excitations in addition to random singlets. Our results signify an intriguing role of quenched disorder in a Kitaev spin system in creating low-energy excitations possibly pertaining to Z$_2$ fluxes.
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