  Effects of the $U(1)_A$ axial anomaly on the mass spectrum of singly heavy
baryons (SHBs) is studied in terms of the chiral effective theory based on the
chiral linear representation for light flavors. We consider SHBs made of both
three quarks ($Qqq$) and five quarks ($Qqqq\bar{q}$). For the three-quark SHBs
we prove that the inverse mass hierarchy for the negative-parity $\Lambda_c$
and $\Xi_c$ is realized only when the $U(1)_A$ anomaly is present. For the
five-quark SHBs, in contrast, it is found that the $U(1)_A$ anomaly does not
change the mass spectrum at the leading order, and accordingly their decay
properties induced by emitting a pseudoscalar meson are not affected by the
anomaly. Moreover, taking into account small mixings between the three-quark
and five-quark SHBs, we find that the observed $\Xi_c$ excited state, either
$\Xi_c(2923)$ or $\Xi_c(2930)$, can be consistently regarded as a
negative-parity SHB that is dominated by the five-quark component. We also
predict a new negative-parity five-quark dominant $\Lambda_c$, whose mass is
around $2700$ MeV and the decay width is of order a few MeV, which provides
useful information for future experiments to check our description.
