  We examine spatial variations of the C$^0$/CO abundance ratio
($X_{\mathrm{C/CO}}$) in the vicinity of the $\gamma$-ray supernova remnant
W51C, based on [CI] ($^3P_1$-$^3P_0$), $^{12}$CO(1-0), and $^{13}$CO(1-0)
observations with the ASTE and Nobeyama 45-m telescopes. We find that
$X_{\mathrm{C/CO}}$ varies in a range of 0.02-0.16 (0.05 in median) inside the
molecular clouds of $A_V>$100 mag, where photodissociation of CO by the
interstellar UV is negligible. Furthermore, $X_{\mathrm{C/CO}}$ is locally
enhanced up to by a factor of four near the W51C center, depending on the
projected distance from the W51C center. In high-$A_V$ molecular clouds,
$X_{\mathrm{C/CO}}$ is determined by the ratio of the cosmic-ray (CR)
ionization rate to the H$_2$ density, and we find no clear spatial variation of
the H$_2$ density against the projected distance. Hence, the high CR ionization
rate may locally enhance $X_{\mathrm{C/CO}}$ near the W51C center. We also find
that the observed spatial extent of the enhanced $X_{\mathrm{C/CO}}$ ($\sim$17
pc) is consistent with the diffusion distance of CRs with the energy of 100
MeV. The fact suggests that the low-energy CRs accelerated in W51C enhance
$X_{\mathrm{C/CO}}$. The CR ionization rate at the $X_{\mathrm{C/CO}}$-enhanced
cloud is estimated to be 3$\times$10$^{-16}$ s$^{-1}$ on the basis of
time-dependent PDR simulations of $X_{\mathrm{C/CO}}$, the value of which is 30
times higher than that in the standard Galactic environment. These results
demonstrate that [CI] is a powerful probe to investigate the interaction
between CRs and the interstellar medium for a wide area in the vicinity of
supernova remnants.
