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We present Atacama Large Millimeter/sub-millimeter Array (ALMA) neutralcarbon, [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 theradio-active galactic nuclei (AGN) host galaxies whose positions are set bynear-infrared detections and radio detections of the cores. Additionally, weplace constraints on the galaxies' systemic redshifts via He II $\lambda$1640lines 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, wediscover narrow line emission of full width at half maximum $\lesssim100$ kms$^{-1}$ which may trace emission from bright kpc-scale gas clouds within theISM. 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 thecold gas. Overall, the [C I] line luminosities correspond to H$_2$ masses ofM$_{\rm H_2,[C I]} \simeq (0.5 - 3) \times 10^{10} M_\odot$ for the detectionsand M$_{H_2,[C I]} < 0.65 \times 10^{10} M_\odot$ for the [C I] non-detectionsin three out of seven galaxies within the sample. The molecular gas masses inour sample are relatively low in comparison to previously reported measures forsimilar galaxies which are M$_{H_2,[C I]} \simeq (3 - 4) \times 10^{10}.$ Ourresults imply that the observed faintness in carbon emission is representativeof a decline in molecular gas supply from previous star-formation epochs and/ora displacement of molecular gas from the ISM due to jet-powered outflows.

Faint [CI](1-0) emission in z $\sim$ 3.5 radio galaxies