There is ever growing demand of modeling overdispersed count data generated by various disiplines. Excessive number of zeros and heterogeneity in the population are two main sources of the overdispersion problem. Development of new count models that are more flexible than conventional Poisson model is thus necessary in order to address such sources. This study fullfils this need by proposing a new heterogeneous Poisson model with a capture of excess zeros, namely zero-inflated Poisson–Ailamujia (ZIPA) model. In line with the aim of curing overdispersion, a censored variant of this newly suggested model is also here developed. An extensive simulation study is conducted to assess the performances of both forms of new models in terms of bias, precision and accuracy measures. Additionally, two real world applications are presented to illustrate practical implications of zero-inflated (censored) Poisson–Ailamujia models in comparison to some alternatives.