We present chemical abundances for the elements carbon, sodium, and fluorine in 15 red giants of the globular cluster M4, as well as six red giants of the globular cluster w Centauri. The chemical abundances were calculated in LTE via spectral synthesis. The spectra analyzed are high-resolution spectra obtained in the near-infrared region around 2.3 mu m with the Phoenix spectrograph on the 8.1 m Gemini South Telescope, the IGRINS spectrograph on the McDonald Observatory 2.7 m Telescope, and the CRIRES spectrograph on the ESO 8.2 m Very Large Telescope. The results indicate a significant reduction in the fluorine abundances when compared to previous values from the literature for M4 and w Centauri, due to a downward revision in the excitation potentials of the HF (1-0) R9 line used in the analysis. The fluorine abundances obtained for the M4 red giants are found to be anticorrelated with those of Na, following the typical pattern of abundance variations seen in globular clusters between distinct stellar populations. In M4, as the Na abundance increases by similar to+0.4 dex, the F abundance decreases by similar to-0.2 dex. A comparison with abundance predictions from two sets of stellar evolution models finds that the models predict somewhat less F depletion (similar to-0.1 dex) for the same increase of +0.4 dex in Na.