We present SPRIGHT, a PYTHON package that implements a fast and lightweight mass-density-radius relation for small planets. The relation represents the joint planetary radius and bulk density probability distribution as a mean posterior predictive distribution of an analytical three-component mixture model. The analytical model, in turn, represents the probability for the planetary bulk density as three generalized Student's t-distributions with radius-dependent weights and means based on theoretical composition models. The approach is based on Bayesian inference and aims to overcome the rigidity of simple parametric mass-radius relations and the danger of overfitting of non-parametric mass-radius relations. The package includes a set of pre-trained and ready-to-use relations based on two M-dwarf catalogues, one catalogue containing stars of spectral types F, G, and K (FGK stars), and two theoretical composition models for water-rich planets. The inference of new models is easy and fast, and the package includes a command line tool that allows for coding-free use of the relation, including the creation of publication-quality plots. Additionally, we study whether the current mass and radius observations of small exoplanets support the presence of a population of water-rich planets positioned between rocky planets and sub-Neptunes. The study is based on Bayesian model comparison and shows somewhat strong support against the existence of a water-world population around M dwarfs. However, the results of the study depend on the chosen theoretical water-world density model. A more conclusive result requires a larger sample of precisely characterized planets and community consensus on a realistic water-world interior structure and atmospheric composition model.