  The gravitational potential of a gas of initially randomly distributed
primordial black holes (PBH) can induce a stochastic gravitational-wave (GW)
background through second-order gravitational effects. This GW background can
be abundantly generated in a cosmic era dominated by ultralight primordial
black holes, with masses $m_\mathrm{PBH}<10^{9}\mathrm{g}$. In this work, we
consider $f(R)$ gravity as the underlying gravitational theory and we study its
effect at the level of the gravitational potential of Poisson distributed
primordial black holes. After a general analysis, we focus on the $R^2$ gravity
model. In particular, by requiring that the scalar induced GWs (SIGWs) are not
overproduced, we find an upper bound on the abundance of PBHs at formation time
$\Omega_\mathrm{PBH,f}$ as a function of their mass, namely that
$\Omega_\mathrm{PBH,f}<5.5\times
10^{-5}\left(\frac{10^9\mathrm{g}}{m_\mathrm{PBH}}\right)^{1/4}$, which is
$45\%$ tighter than the respective upper bound in general relativity.
Afterwards, by considering $R^2$ gravity as an illustrative case study of an
$f(R)$ gravity model, we also set upper bound constraints on its mass parameter
$M$. These mass parameter constraints, however, should not be regarded as
physical given the fact that the Cosmic Microwave Background (CMB) constraints
on $R^2$ gravity are quite tight. Finally, we conclude that the portal of SIGWs
associated to PBH Poisson fluctuations can act as a novel complementary probe
to constrain alternative gravity theories.
