A resolvent theoretical analysis of the cyclotron-resonance (CR) line shape of a two-dimensional hole gas subjected to a high magnetic field applied perpendicularly to the hole layer in microstructures such as the heterojunctions, single quantum wells, or multiple-quantum-well structures is developed by considering the inelastic hole-phonon scattering in the effective-mass approximation along the lines developed earlier [Phys. Rev. B 34, 3932 (1986)]. Divergence in the power absorption spectra is removed by renormalizing the initial and final single-hole states due the phonons which is equivalent to solving for the matrix elements of the tetradic self-energy function in the self-consistent Born approximation. General coupled equations for the CR frequency shifts (${\ensuremath{\Delta}}_{N}$,${\mathit{p}}^{\mathrm{M}}$) and CR linewidths (${\ensuremath{\Gamma}}_{N}$,${\mathit{p}}^{\mathrm{M}}$) for the heavy (M=\ifmmode\pm\else\textpm\fi{}3/2) and light (M=\ifmmode\pm\else\textpm\fi{}1/2) holes are derived. Approximate solution of the derived equations is discussed in the limit of high magnetic field and elastic hole-phonon scattering for the deformation-potential acoustic and piezoelectric hole-phonon interaction in a two-dimensional phonon model and neglecting the coupling between the heavy- and light-hole bands. In this simplistic approximation the CR frequency shifts for heavy and light holes are found to vanish (${\mathrm{\ensuremath{\Delta}}}_{\mathit{N},\mathit{p}}^{\ifmmode\pm\else\textpm\fi{}3/2}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{N},\mathit{p}}^{\ifmmode\pm\else\textpm\fi{}1/2}$=0) and CR linewidths increase with magnetic field for the deformation-potential acoustic- and piezoelectric-phonon scattering as ${B}^{1/2}$ and ${B}^{1/4}$, respectively, whereas the temperature variation for both the scattering mechanisms is given as ${T}^{1/2}$.