  A procedure of testing the $f(R)$-theory of gravity is discussed. The latter
is an extension of the general theory of relativity (GR). In order this
extended theory (in some variant) to be really confirmed as a more precise
theory it must be tested. To do that we first have to solve an equation
generalizing Einstein's equation in the GR. However, solving this generalized
Einstein's equation is often very hard, even it is impossible in general to
find an exact solution. It is why the perturbation method for solving this
equation is used. In a recent work \cite{Ky:2018fer} a perturbation method was
applied to the $f(R)$-theory of gravity in a central gravitational field which
is a good approximation in many circumstances. There, perturbative solutions
were found for a general form and some special forms of $f(R)$. These solutions
may allow us to test an $f(R)$-theory of gravity by calculating some quantities
which can be verified later by the experiment (observation). In
\cite{Ky:2018fer} an illustration was made on the case $f(R)=R+\lambda R^2$.
For this case, in the present article, the orbital precession of S2 orbiting
around Sgr A* is calculated in a higher-order of approximation. The
$f(R)$-theory of gravity should be also tested for other variants of $f(R)$ not
considered yet in \cite{Ky:2018fer}. Here, several representative variants are
considered and in each case the orbital precession is calculated for the
Sun--Mercury- and the Sgr A*--S2 gravitational systems so that it can be
compared with the value observed by a (future) experiment. Following the same
method of \cite{Ky:2018fer} a light bending angle for an $f(R)$ model in a
central gravitational field can be also calculated and it could be a useful
exercise.
