The high-fidelity measurement of the ultra-short and ultra-intense laser pulses' temporal signal-to-noise ratio (SNR) is of great significance. To the best of our knowledge, few studies have investigated the influence of wavefront distortion on the measurement of pulse SNR. In this work, a numerical model is constructed to study how wavefront distortion affects the measurement of ultra-short and ultra-intense pulse SNR by the single-shot third-order auto-correlation (TOAC) method. The nonlinear coupled-wave equations with wavefront distortion have been solved numerically by the split-step Fourier method and the fourth-order Runge-Kutta numerical algorithm. The wavefront distortion of the under-test fundamental wave will be transmitted to the second harmonic and third harmonic, leading to the phase mismatch in the second harmonic generation (SHG) and third harmonic generation (THG), further resulting in the deterioration of the measured SNR. We analyze the influence of different spatial frequencies and peak-to-valley (PV) values on the measurements of SNR. The larger the spatial frequency or PV value of the wavefront distortion, the more severe the degradation of the SNR.