The presence of strong anharmonic effects in surface functionalized MXenes greatly challenges the use of harmonic lattice dynamics calculations to predict their phonon spectra and lattice thermal conductivity at finite temperatures. Herein, we demonstrate the workflow for training and validating machine learning potentials in terms of moment tensor potential (MTP) for MXenes including Mo₂TiC₂, Mo₂TiC₂O₂, Mo₂TiC₂F₂ and Janus-Mo₂TiC₂OF monolayers. Then, the MTPs of MXenes are successfully combined with the harmonic lattice dynamics calculations to obtain the temperature renormalized phonon spectra, three-phonon scattering rates, phonon relaxation times and lattice thermal conductivity at finite temperatures. Furthermore, combining MTPs with classic molecular dynamics simulations at finite temperatures directly enables the calculation of phonon quasi-particle spectral energy density with a full inclusion of all anharmonic effects in MXenes. Our current results indicate that anharmonic effects are found to be relatively weak in Mo₂TiC₂ and Mo₂TiC₂O₂ monolayers, whereas the phonon quasi-particle spectral energy densities largely resemble those of harmonic or renormalized lattice dynamics calculations. Significant broadening of spectral energy density at finite temperature is predicted for Mo₂TiC₂F₂ and Janus-Mo₂TiC₂OF monolayers, implying strong anharmonic effects in those MXenes. Our work paves a new way for fast and reliable calculation of the phonon scattering process and lattice thermal conductivity of MXenes within MTPs trained from first-principles molecular dynamics simulations in the future.