A new cure rate model is presented by assuming that, conditional on η, the number of competing causes of the event of interest follows the Poisson distribution, where η is assumed a random variable with gamma and generalized exponential distributions. For the time-to-event of the concurrent causes, we assumed the recently introduced truncated Nadarajah-Haghighi model. The model is parameterized directly in terms of the cure term, and then different symmetric and asymmetric link functions are used to assess the effects of covariates, such as logit, probit, log-log, Cauchit, Aranda-Ordaz, skewed probit, and skewed logit. Parameters estimation for the model is approached based on the traditional maximum likelihood estimation method. We achieve a simulation study in order to investigate the performance of these estimators under different scenarios. Finally, the model is illustrated in data sets related to two kinds of cancers (melanoma and colon cancer).