Hexaferrite materials are highly demanded to develop and manufacture electronic devices operating at radio- and microwave frequencies. In the light of the prospects for their use in the forthcoming terahertz electronics, here, we present our results on the terahertz and infrared dielectric response of a typical representative of hexaferrites family, lead-substituted M-type barium hexaferrite doped with aluminum, Ba0.2Pb0.8AlxFe12-xO19, x(Al)=0.0, 3.0, and 3.3. We studied uniquely large and high-quality single crystals of the. Systematic and detailed investigations of the dependences of terahertz-infrared (frequencies 8 - 8000 cm-1) spectra of complex dielectric permittivity on the temperature, 4 - 300 K, and on the chemical composition, x(Al)=0.0, 1.2, 3.0, 3.3, were performed for polarizations of the electric field E-vector of the probing radiation normal and parallel to the crystallographic c-axis. A number of resonance absorption bands are discovered at infrared-terahertz frequencies and assigned to polar phonons and transitions between energy levels of the fine-structured ground state of Fe2+ (5E) ions. In contrast to undoped BaFe12O19, no softening of the lowest frequency A2u phonon is observed, indicating suppression of a displacive phase transition in substituted compounds. Basing on dielectric data and detailed X-ray experiments, we find that for all concentrations of Al3+ ions, x(Al)=0.0, 1.2, 3.0, and 3.3, they mainly occupy the 2a and 12k octahedral site positions and that the degree of substitution of iron in tetrahedral positions is not substantial. Along with fundamental findings, the obtained data on broad-band dielectric properties of Ba0.2Pb0.8AlxFe12-xO19 crystals provides the information that can be used for development and manufacture of electronic devices with operating frequencies lying in the terahertz spectral band.