  To clarify the origin of the nontrivial thermoelectric properties observed in
the delafossite oxide CuRhO$_2$ polycrystals, we have performed the systematic
transport measurements on the single-crystalline CuRhO$_2$ samples. In the
parent compound, we find a pronounced peak structure due to a phonon-drag
effect in the temperature dependence of the Seebeck coefficient, which is also
confirmed by the size effect experiments. In the Mg-substituted crystals, in
contrast to the results of the polycrystals, both the resistivity and the
Seebeck coefficient decrease with increasing Mg content $y$. In particular, the
coefficient $A$ for the $T^2$ term of the resistivity and the $T$-linear
coefficient for the Seebeck coefficient at low temperatures are well described
within a simple relationship expected for metals, which is also applicable to
the correlated materials with low carrier densities.
