The strong-field approximation, amended so as to allow for rescattering, is used to calculate high-order above-threshold ionization (ATI) spectra. The single-active-electron binding potential is modelled by a zero-range potential. The emphasis is on enhancements of groups of ATI peaks that occur for sharply defined laser intensities. The enhancements are traced to multiphoton resonance with the ponderomotively upshifted continuum threshold. Good agreement both with experimental data and with numerical simulations using the three-dimensional time-dependent Schrodinger equation for an optimized one- electron binding potential is observed. The physical reason for the close agreement of the results of the two, apparently so different, models is discussed. For quantitative agreement with the experimentally observed positions of the resonances, an 'effective' continuum threshold has to be introduced. The effects of focal averaging are evaluated and discussed. Resonant enhancement of high-order harmonic generation is also considered.