  The linear perturbation of the Kerr black hole has been discussed by using
the Newman--Penrose and the perturbed Weyl scalars, $\psi_0$ and $\psi_4$ can
be obtained from the Teukolsky equation. In order to obtain the other Weyl
scalars and the perturbed metric, a formalism was proposed by Chrzanowski and
by Cohen and Kegeles (CCK) to construct these quantities in a radiation gauge
via the Hertz potential. As a simple example of the construction of the
perturbed gravitational field with this formalism, we consider the
gravitational field produced by a rotating circular ring around a Schwarzschild
black hole. In the CCK method, the metric is constructed in a radiation gauge
via the Hertz potential, which is obtained from the solution of the Teukolsky
equation. Since the solutions $\psi_0$ and $\psi_4$ of the Teukolsky equations
are spin-2 quantities, the Hertz potential is determined up to its monopole and
dipole modes. Without these lower modes, the constructed metric and
Newman--Penrose Weyl scalars have unphysical jumps on the spherical surface at
the radius of the ring. We find that the jumps of the imaginary parts of the
Weyl scalars are cancelled when we add the angular momentum perturbation to the
Hertz potential. Finally, by adding the mass perturbation and choosing the
parameters which are related to the gauge freedom, we obtain the perturbed
gravitational field which is smooth except on the equatorial plane outside the
ring. We discuss the implication of these results to the problem of the
computation of the gravitational self-force to the point particles in a
radiation gauge.
