Clustered survival data frequently occur in medical research, especially in clinical trials and cohort studies. Observations within clusters may be correlated due to some natural, artificial clustering or shared environmental factors of subjects that may influence the failure times of the same cluster. We propose a Weibull regression model with frailty, generated by a flexible distribution, which provides a better description of the dependency structure on the data. The Weibull regression model can be considered as an attractive alternative to the Cox proportional hazard model in analyzing survival data.Furthermore, we consider a log-skew-normal distribution of the frailty, leading to an extension of the log-normal frailty model. Complex multidimensional integrals are avoided by using hierarchical likelihood (h-likelihood) to estimate the regression parameters and to predict the realizations of random effects. The adjusted profile hierarchical likelihood is adopted to estimate the parameters in frailty distribution. We illustrate our method using a kidney catheter dataset and Monte-Carlo simulation.