Count data play a critical role in medical research, such as heart disease. The Poisson regression model is a common technique for evaluating the impact of a set of covariates on the count responses. The mixture of Poisson regression models with experts is a practical tool to exploit the covariates, not only to handle the heterogeneity in the Poisson regressions but also to learn the mixing structure of the population. Multicollinearity is one of the most common challenges with regression models, leading to ill-conditioned design matrices of Poisson regression components and expert classes. The maximum likelihood method produces unreliable and misleading estimates for the effects of the covariates in multicollinearity. In this research, we develop Ridge and Liu-type methods as two shrinkage approaches to cope with the ill-conditioned design matrices of the mixture of Poisson regression models with experts. Through various numerical studies, we demonstrate that the shrinkage methods offer more reliable estimates for the coefficients of the mixture model in multicollinearity while maintaining the classification performance of the ML method. The shrinkage methods are finally applied to a heart study to analyze the heart disease rate stages.