  We present Atacama Large Millimeter/sub-millimeter Array (ALMA) neutral
carbon, [C I](1-0), line observations that probe molecular hydrogen gas (H$_2$)
within seven radio galaxies at $z = 2.9 - 4.5$ surrounded by extended
($\gtrsim100$ kpc) Ly-$\alpha$ nebulae. We extract [C I](1-0) emission from the
radio-active galactic nuclei (AGN) host galaxies whose positions are set by
near-infrared detections and radio detections of the cores. Additionally, we
place constraints on the galaxies' systemic redshifts via He II $\lambda$1640
lines seen with the Multi-Unit Spectroscopic Explorer (MUSE). We detect faint
[C I] emission in four out of seven sources. In two of these galaxies, we
discover narrow line emission of full width at half maximum $\lesssim100$ km
s$^{-1}$ which may trace emission from bright kpc-scale gas clouds within the
ISM. In the other two [C I]-detected galaxies, line dispersions range from
$\sim100 - 600$ km s$^{-1}$ and may be tracing the rotational component of the
cold gas. Overall, the [C I] line luminosities correspond to H$_2$ masses of
M$_{\rm H_2,[C I]} \simeq (0.5 - 3) \times 10^{10} M_\odot$ for the detections
and M$_{H_2,[C I]} < 0.65 \times 10^{10} M_\odot$ for the [C I] non-detections
in three out of seven galaxies within the sample. The molecular gas masses in
our sample are relatively low in comparison to previously reported measures for
similar galaxies which are M$_{H_2,[C I]} \simeq (3 - 4) \times 10^{10}.$ Our
results imply that the observed faintness in carbon emission is representative
of a decline in molecular gas supply from previous star-formation epochs and/or
a displacement of molecular gas from the ISM due to jet-powered outflows.
