We propose a theory of the electron-phonon drag effect in an ideally pure 2D system under a quantizing magnetic field. Two situations are considered, i.e. when phonons are (i) those of a diffusive heat flow and (ii) those of a focused ballistic beam. In the first case, we calculate the drag current for different asymptotical temperature intervals and give its relation to the phonon-scattering-induced conductivity. For the 2D gas illuminated by a hot beam of phonons, we analyse the manifestation of their cyclotron absorption both in the drag and in the beam-activated conductivity.