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Recently, Mlaiki et al. introduced the notion of a controlled metric type space which is a generalization of the <i>b</i>-metric space. In this work, we define the controlled partial metric type space and give some fixed-point theorems for extensions of Kannan contraction in this space with suitable conditions. Moreover, as an application, we derive a fixed-point theorem for graphic contraction on the considered metric space endowed with a graph.