Time-to-event data analysis has a long tradition in applied statistics. Many models have been developed for data where each subject or observation unit experiences at most one event during its life. In contrast, in some applications, the subjects may experience more than one event. Recurrent events appear in science, medicine, economy, and technology. Often the events are followed by a repair action in reliability or a treatment in life science. A model to deal with recurrent event times for incomplete repair of technical systems is the trend-renewal process. It is composed of a trend and a renewal component. In the present paper, we use a Weibull process for both of these components. The model is extended to include a Cox type covariate term to account for observed heterogeneity. A further extension includes random effects to account for unobserved heterogeneity. We fit the suggested version of the trend-renewal process to a data set of hospital readmission times of colon cancer patients to illustrate the method for application to clinical data.