  A modified gravity model of Starobinsky inflation and primordial black hole
production was proposed in good (within $1\sigma$) agreement with current
measurements of the cosmic microwave background radiation. The model is an
extension of the singularity-free Appleby-Battye-Starobinsky model by the
$R^4$-term with different values of the parameters whose fine-tuning leads to
efficient production of primordial black holes on smaller scales with the
asteroid-size masses between $10^{16}$ g and $10^{20}$ g. Those primordial
black holes may be part (or the whole) of the current dark matter, while the
proposed model can be confirmed or falsified by detection or absence of the
induced gravitational waves with the frequencies about $10^{-2}$ Hz. The main
purpose of this paper was to estimate the size of quantum (loop) corrections in
the model. It was found their relative contribution to the power spectrum of
scalar perturbations is about $10^{-3}$ or less, so that the model is not ruled
out by the quantum corrections.
