In order to complement the randomization of the Eigen model and the Crow-Kimura model, the site mutation rate in the Crow-Kimura model is treated as a Gaussian distributed random variable. The characteristics of the error threshold as well as the relationship between the extension of the error threshold and the fluctuation strength of the randomized mutation rate are investigated. It is shown that both the relative concentrations and the order parameter indicate the error threshold is no longer a phase transition point but a smooth crossover region in the presence of a sizable fluctuation of the site mutation rate. The quantitative analysis demonstrates that the relationship between the width of the crossover region and the fluctuation strength in the mutation randomized Crow-Kimura model is nonlinear. The obtained results are compared with those from the randomized Eigen model and it is found that for the two randomized models the relationship between the width and the fluctuation strength is linear for the randomized fitness and nonlinear (exponential) for the randomized site mutation rate. For the randomized Crow-Kimura model the width caused by the randomized fitness is comparable to that caused by the randomized site mutation rate. Nevertheless, for the randomized Eigen model the width is mainly caused by the randomized site mutation rate. A full picture about the randomization effects of the fitness and site mutation rate on the error threshold based of the Eigen model and the Crow-Kimura model is then outlined. The implications of the above results for anti-viral strategies, cancer therapy and breeding of animals and plants are discussed.