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A formulation for the high-order harmonic generation (HHG) amplitude [M. V. Frolov et al., Phys. Rev. A 75, 063407 (2007), preceding paper] is employed to provide analytical results for HHG rates within our recently developed time-dependent effective range (TDER) theory (for time-dependent problems involving weakly bound electron systems). Exact and approximate (including quasiclassical) TDER HHG rates are employed to analyze the accuracy of common approximate methods for HHG calculations. For various specific negative ions with s and p outer electrons, numerical results for HHG spectra are presented over a wide interval of laser frequencies(extending from the tunneling to the multiphoton regimes). The role of initial bound state symmetry effects on the HHG spectra is also analyzed. Finally, Coulomb corrections to TDER results for HHG rates are introduced and discussed.