GaAs/Al0.33Ga0.67As quantum-cascade lasers with plasmon-assisted waveguides exhibit a decreasing threshold current density jth with increasing wave number ν0 of the laser line, which changes as a function of the injector doping density. We have developed an analytical approach based on the effective dielectric tensor component for the p-polarized light emitted from a quantum-cascade laser, which explains the observed dependence of jth(ν0) in terms of losses due to free-carrier absorption predominantly in the doped waveguides αWG(ν0). A contribution to the losses by free-carrier absorption in the quantum-cascade structure itself and subsequently to jth can be neglected except for very high injector doping densities. The calculated values for αWG(ν0) are in good agreement with the experimental data. Our approach quantitatively predicts the observed decrease of jth from 17 to 7 kA cm-2 with increasing ν0 between 900 and 1100 cm-1. In addition to achieving a direct physical insight into the influence of free-carrier absorption on the laser performance, the proposed analytical approach provides a simple tool for the determination of the waveguide losses for any quantum-cascade laser without adopting a numerical solver.