We present new, spatially resolved [CI]1-0, [CI]2-1, CO(7-6), and dust continuum observations of 4C 41.17 at $z=3.8$ obtained with the IRAM NOEMA interferometer. This is one of the best-studied radio galaxies in this epoch and is arguably the best candidate of jet-triggered star formation at high redshift currently known in the literature. 4C 41.17 shows a narrow ridge of dust continuum extending over 15 kpc near the radio jet axis. Line emission is found within the galaxy in the region with signatures of positive feedback. Using the [CI]1-0 line as a molecular gas tracer, and multifrequency observations of the far-infrared dust heated by star formation, we find a total gas mass of $7.6times 10^{10}$ M$_{odot}$, which is somewhat greater than that previously found from CO(4-3). The gas mass surface density of $10^3$ M$_{odot}$ yr$^{-1}$ pc$^{-2}$ and the star formation rate surface density of 10 M$_{odot}$ yr$^{-1}$ kpc$^{-2}$ were derived over the 12 kpc$times$8 kpc area, where signatures of positive feedback have previously been found. These densities are comparable to those in other populations of massive, dusty star-forming galaxies in this redshift range, suggesting that the jet does not currently enhance the efficiency with which stars form from the gas. This is consistent with expectations from simulations, whereby radio jets may facilitate the onset of star formation in galaxies without boosting its efficiency over longer timescales, in particular after the jet has broken out of the interstellar medium, as is the case in 4C 41.17.
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