  We argue that the large suppression of the $\psi(2S)$ inclusive cross-section
relative to the $J/\psi$ inclusive cross-section in proton-nucleus (p+A)
collisions can be attributed to factorization breaking effects in the formation
of quarkonium. These factorization breaking effects arise from soft color
exchanges between charm-anticharm pairs undergoing hadronization and comoving
partons that are long-lived on time scales of quarkonium formation. We compute
the short distance pair production of heavy quarks in the Color Glass
Condensate (CGC) effective field theory and employ an improved Color
Evaporation model (ICEM) to describe their hadronization into quarkonium at
large distances. The combined CGC+ICEM model provides a quantitative
description of $J/\psi$ and $\psi(2S)$ data in proton-proton (p+p) collisions
from both RHIC and the LHC. Factorization breaking effects in hadronization,
due to additional parton comovers in the nucleus, are introduced heuristically
by imposing a cutoff $\Lambda$, representing the momentum kick from soft color
exchanges, in the ICEM model. Such soft exchanges have no perceptible effect on
$J/\psi$ suppression in p+A collisions. In contrast, the interplay of the
physics of these soft exchanges at large distances, with the physics of
semi-hard rescattering at short distances, causes a significant additional
suppression of $\psi(2S)$ yields relative to that of the $J/\psi$. A good fit
of all RHIC and LHC $J/\psi$ and $\psi(2S)$ data, for transverse momenta
$P_\perp\leq 5$ GeV in p+p and p+A collisions, is obtained for $\Lambda\sim 10$
MeV.
