  In a recent paper, Nashed and Capozziello presented a new class of charged,
spherically symmetric black hole solutions of $f(R)$ gravity with an asymptotic
flat or (anti-)de Sitter behavior. These metrics depend on a dimensional
parameter $\alpha$ and are interesting because they cannot reduce to general
relativity solutions. In this paper, we present a corrected study of their
physical and thermodynamic properties and generalize these solutions to obtain
a new set of stationary, axisymmetric black holes in the $f(R)$ scenario. Some
of our results show that the entropy is always positive within the allowed
values of parameter $\alpha$ and due to the well behaved quantities such as the
Gibbs free energy, we conclude that there is no such a phase transition as
discussed in the work of Nashed and Capozziello. We also study the geodesics in
these spacetimes and particularly, the stability of the circular orbits to
obtain the radius of the Innermost Stable Circular Orbit.
