While the Bartlett-Lewis model has been widely used for modelling rainfall processes at a fixed point in space over time, there are observed features, such as longer-scale dependence, which are not well fitted by the model. In this paper, we study an extension where we put an extra layer in the clustered Poisson process of storm origins. We also investigate the Pareto inter-arrival time for the storm origins, which has been used to model web-traffic data. We derive the theoretical first and second-order properties of the multi-layer clustered Poisson processes, but generally we have to rely on Monte Carlo techniques. The models are fitted to hourly rainfall data from Valentia observatory in southwest Ireland, where the extensions are shown to improve on the standard models. We generalize these models further by allowing some parameters of the models to be a function of some covariates. An application using data from Valentia observatory and Belmullet shows how to use this class of models to analyze the association between the rainfall pattern and the North Atlantic Oscillation (NAO) index.