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¹⁹F NMR spectra and spin-lattice relaxation rates have been measured on two powder HgBa₂CuO₄Fₓ samples. The results combined with our neutron powder-diffraction data [A. M. Abakumov et al., Phys. Rev. Lett. 80, 385 (1998)] confirm that fluorine atoms can be successfully incorporated in the lattice of superconducting Hg-1201 phase in the nonstoichiometric O3 site. Partial occupation of another oxygen position by fluorine atoms reveals to be possible with the increase of F content. The temperature dependence of the ¹⁹F NMR linewidth reveals the crossover at 35 K in the magnetic-flux line structure between the low-temperature latticelike regime and high-temperature “melted” regime. For the 1/T₁ temperature dependence a maximum has been found at 65 K indicating the strong influence of magnetic-flux line motion on the ¹⁹F spin-lattice relaxation rate.