Strained mercury telluride (HgTe) thin films with the critical thickness of 6.5 nm possess a gapless Dirac-like band structure. We report a comprehensive study of both gated and optically doped samples by the means of magnetooptical spectroscopy in the THz range. Analysis of the spectra and magnetic field dependencies at fixed frequencies within the Drude model has allowed to map the band dispersion of the Dirac charge carriers in the broad range of densities from 1 to 80·1010 cm-2 and the cyclotron mass range from 0.001 to 0.03 of the free electron mass (m0). A smooth transition through the charge neutrality point between Dirac holes and electrons was observed. Unexpectedly, an additional peak coming from another type of holes with the almost density independent mass of around 0.04m0 was detected in the hole doping range as well. The response of electron-like Dirac carriers at higher densities exhibit clear deviations from the Drude model in the form of Shubnikov-de Haas-like oscillations. This all-optical high frequency (0.95 THz) detection of the quantum corrections to the AC conductivity excludes contact or edge based theoretical models.