  Infinite-layer nickelates ($R$NiO$_2$) exhibit some distinct differences as
compared to cuprate superconductors, leading to a debate concerning the role of
rare-earth ions ($R$=La, Pr, Nd) in the low-energy many-body physics. Although
rare-earth $4f$ orbitals are typically treated as inert `core' electrons in
studies, this approximation has been questioned. An active participation of
$4f$ states is most likely for PrNiO$_2$ based on an analogy to cuprates where
Pr cuprates differ significantly from other cuprates. Here, we adopt density
functional plus dynamical mean field theory (DFT+DMFT) to investigate the role
of Pr $4f$ orbitals and more generally the correlated electronic structure of
PrNiO$_2$ and its hole-doped variant. We find that the Pr $4f$ states are
insulating and show no evidence for either a Kondo resonance or Zhang-Rice
singlet formation as they do not have any hybridization channels near the Fermi
energy. The biggest effects of hole doping are to shift the Pr $5d$ and $4f$
states further away from the Fermi energy while enhancing the Ni $3d$ - O $2p$
hybridization, thus reducing correlation effects as the O $2p$ states get
closer to the Fermi energy. We again find no evidence for either Kondo or
Zhang-Rice physics for the $4f$ states upon hole doping. We conclude by
commenting on implications for other reduced valence nickelates.
