  We investigate a model of $R^2$-gravity with a non-minimally coupled scalar
field that gives rise to two-stage inflation with a break, that is, with an
intermediate stage where inflation momentarily halts. We find that the power
spectrum of the primordial curvature perturbation is significantly enhanced at
the break scale, which can account for the primordial black hole (PBH)
formation, without affecting the CMB constraint on large scales. The behavior
of the curvature perturbation is carefully analyzed and we find a few notable
new features in the spectrum. In particular, we find that the $k^3$ growth of
the spectrum of toward the end of the first stage of inflation. We argue that
this is a universal feature common to all two-stage models where the field
dominating the second stage is heavy during the first stage. By appropriately
tuning the model parameters, we find that our model can realize the scenario of
PBHs as the cold dark matter of the Universe. We also find that we can choose
the parameters so that the spectrum of the induced gravitational waves from the
enhanced curvature perturbation fits the NANOGrav-15yr data of pulsar timing
array observation.
