In this paper, the Kubo-Anderson process has been adopted to discribe the random frequency modulation of driven field. Its suppression effect on inhomogeneous cold plasma wavebreaking has been analysed. The expressions of wavebreaking time, fast electron energy and the average square of maximum localized electric field have been derived, which can be applied in any band width of the frequency modulation while the Bogoliubov's nonlinear frequency shift is not taken into account. The agreement between the theoretical predictions and the experimental results is fair. Thus we show that the random frequency modulation is an effective method to suppress the wavebreaking and fast electron energy.