Recently, Hennebelle and Chabrier derived an analytical theory for the mass spectrum of non-self-gravitating clumps associated with overdensities in molecular clouds and for the initial mass function (IMF) of gravitationally bound prestellar cores, as produced by the turbulent collapse of the cloud. In this companion paper, we examine the effects of the nonisothermality of the flow, of the turbulence forcing and of local fluctuation of the velocity dispersion, on the mass function. In particular, we investigate the influence of a polytropic equation of state (eos) and of the effective adiabatic exponent gamma and find that it has a drastic influence on the low-mass part of the IMF. We also consider a barotropic eos (i.e., a piecewise polytropic eos) that mimics the thermal behavior of the molecular gas and compare the prediction of our theory with the results of numerical simulations and with the observationally derived IMF, for cloud parameters which satisfy Larson-type relations. We find that for clouds whose density is, at all scales, almost an order of magnitude larger than the density inferred for the CO clumps in the Galaxy, a good agreement is obtained between the theory and the observed IMF, suggesting that star formation preferentially occurs in high-density environments. We derive an analytical expression for the IMF which generalizes the expression previously obtained for the isothermal case. This easy-to-implement analytical IMF should serve as a template to compare observational or numerical results with the theory.