ABSTRACT We present Atacama Large Millimeter/sub-millimeter Array neutral carbon, [${\rm C\, \small {\rm I}}$](1–0), line observations that probe molecular hydrogen gas (H2) within seven radio galaxies at z = 2.9−4.5 surrounded by extended (≳100 kpc) Ly α nebulae. We extract [${\rm C\, \small {\rm I}}$](1–0) emission from the radio-active galactic nuclei (AGNs) 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 ${\rm He\, \small {\rm II}}$ λ1640 lines seen with the Multi-Unit Spectroscopic Explorer. We detect faint [${\rm C\, \small {\rm I}}$] emission in four out of seven sources. In two of these galaxies, we discover narrow-line emission of full width at half-maximum ≲100 km s−1 that may trace emission from bright kpc-scale gas clouds within the interstellar medium (ISM). In the other two [${\rm C\, \small {\rm I}}$]-detected galaxies, line dispersions range from ∼100−600 km s−1 and may be tracing the rotational component of the cold gas. Overall, the [${\rm C\, \small {\rm I}}$] line luminosities correspond to H2 masses of $M_{\rm H_2,[{\rm C\, \small {\rm I}}]} ≃ (0.5{\!-\!}3) \times 10^{10}$ M⊙ for the detections and $M_{\rm H_2,[{\rm C\, \small {\rm I}}]}\lt 0.65\times 10^{10}$ M⊙ for the [${\rm C\, \small {\rm I}}$] 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 that are $M_{\rm H_2,[{\rm C\, \small {\rm I}}]} ≃ (3{\!-\!}4) \times 10^{10}$ M⊙. 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.