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In this paper we introduce the notion of strongly Lipschitz weakly p-nuclear from a pointed metric space X into a Banach space E. We show that they can be seen as a natural extension of the linear weakly p-nuclear operators, we transfer some properties of the linear case into the Lipschitz setting and we introduced this concept for X and E are pointed metric spaces (we say that ”Lipschitz weakly p-nuclear operators”) . In addition, we introduce the ideal of Lipschitz weakly quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz p-nuclear operators.