  We analyze the properties of impurities immersed in a vortex lattice formed
by ultracold bosons in the mean field quantum Hall regime. In addition to the
effects of a periodic lattice potential, the impurity is dressed by collective
modes with parabolic dispersion (Tkachenko modes). We derive the effective
polaron model, which contains a marginal impurity-phonon interaction. The
polaron spectral function exhibits a Lorentzian broadening for arbitrarily
small wave vectors even at zero temperature, in contrast with the result for
optical or acoustic phonons. The anomalous damping of Tkachenko polarons could
be detected experimentally using momentum-resolved spectroscopy.
