  $f(R)$ gravity is one of the serious alternatives of general relativity
having large range of astronomical consequences. In this work, we study Big
Bang Nucleosynthesis (BBN) in $f(R)$ gravity theory. We consider modification
to gravity due to the existence of primordial black holes in the radiation era
which introduce additional degrees of freedom known as scalarons. We calculate
the light element abundances by using the BBN code PArthENoPE. It is found that
for a range of scalaron mass $(2.2-3.5) \times 10^4$ eV, the abundance of
lithium is lowered by $3-4$ times the value predicted by general relativistic
BBN which is a level desired to address the cosmological lithium problem. For
the above scalaron mass range helium abundance is within observed bound.
However, deuterium abundance is found to be increased by $3-6$ times the
observed primordial abundance which calls for high efficiency of stellar
formation and evolution processes for destruction of the same. A novel relation
between scalaron mass and black hole mass has been used to estimate that the
above scalaron mass range corresponds to primordial black holes of
sub-planetary mass ($\sim 10^{19}$ g) serving as one of the potential
non-baryonic dark matter candidates. We infer Big Bang equivalence of power law
$f(R)$ gravity with primordial black holes.
