  We investigate instabilities of $F(R)$-charged $AdS_{4}$ black holes by a
massive charged scalar field in a linear perturbation regime. We study
tachyonic instabilities as the near horizon scalar condensation in a model of
$F(R)$ gravity with planar horizon and investigate properties of possible phase
transitions. The results show that such transitions are sensitive to the first
derivative of $F(R)$ with respect to $R$ in that the larger its value, the
higher the critical temperature, thus resulting in a new generation of
high-temperature superconductors. Also, for a certain range of parameters,
$F(R)$-charged $AdS_{4}$ black holes suffer from superradiant instability. We
consider the effects of the scalar mass and charge on such instabilities and
conclude that RN black holes decay into small hairy black holes that have a
charged scalar condensate floating near the horizon. It is shown that the
existence of phase transition at the critical temperature leading to a hairy
black hole solution emerges for $T<T_{c}$, while RN black holes exist for
$T>T_{c}$. The effect of $F(R)$ on the critical temperature is subsequently
investigated in the case of superradiant instability, showing that higher
critical temperatures would be possible in $F(R)$ gravity. We also check the
stability of hairy black holes and show that the resulting hairy solution can
be considered as a possible end point of superradiant instability of a small
charged black hole.
