This paper introduces and studies a new discrete distribution with one parameter that expands the Poisson model, discrete weighted Poisson Lerch transcendental (DWPLT) distribution. Its mathematical and statistical structure showed that some of the basic characteristics and features of the DWPLT model include probability mass function, the hazard rate function for single and double components, moments with auxiliary statistical measures (expectation, variance, index of dispersion, skewness, kurtosis, negative moments), conditional expectation, Lorenz function, and order statistics, which were derived as closed forms. DWPLT distribution can be used as a flexible statistical approach to analyze and discuss real asymmetric leptokurtic data. Moreover, it could be applied to a hyperdispersive data model. Two different estimation methods were derived, i.e., maximal likelihood and the moments technique for the DWPLT parameter, and some advanced numerical methods were utilized for the estimation process. A simulation was performed to examine and analyze the performance of the DWPLT estimator on the basis of the criteria of the bias and mean squared errors. The flexibility and fit ability of the proposed distribution is demonstrated via the clinical application of a real dataset. The DWPLT model was more flexible and worked well for modeling real age data when compared to other competitive age distributions in the statistical literature.