Fitting mixture regression models has received attention from the authors in the statistical literature over the last decade. That is due to its ability to model data with more than one pattern. The main reason is the latent variables, which are the source of these patterns appearing. The EM algorithm is considered the enhanced approach to the MLE estimation method for estimating its parameters. However, this method is not resistant to the outliers; heavy-tailed distributions are recommended to fit the mixture regression model. Fitting a regression model with a Laplace mixture distribution was proposed in the literature to be robust against outliers and leverage points. Whatever, the previous studies sought to trim the leverage points, while this manuscript seeks to weight the leverage points instead of trimming them. We noted that the mixture model has four types of outliers, which are considered in the simulation study. The performance of the proposed method, W.MixLap, is compared with MixLap, mixt, and MLE, which are fitted regression models with Laplace, Student, and Gaussian mixture distributions, respectively. The result shows that the performance of W.MixLap is better than others.