Kidney cancer, a potentially life-threatening malignancy affecting the kidneys, demands early detection and proactive intervention to enhance prognosis and survival. Advancements in medical and health sciences and the emergence of novel treatments are expected to lead to a favorable response in a subset of patients. This, in turn, is anticipated to enhance overall survival and disease-free survival rates. Cure fraction models have become essential for estimating the proportion of individuals considered cured and free from adverse events. This article presents a novel piecewise power-law cure fraction model with a piecewise decreasing hazard function, deviating from the traditional piecewise constant hazard assumption. By analyzing real medical data, we evaluate various factors to explain the survival of individuals. Consistently, positive outcomes are observed, affirming the significant potential of our approach. Furthermore, we use a local influence analysis to detect potentially influential individuals and perform a postdeletion analysis to analyze their impact on our inferences.