In this paper we introduce a variable order time fractional differential equation driven by pure jump Lévy noise, which models the motion of a particle exhibiting memory effect. We prove the well-posedness of this equation without assuming any integrability condition on the initial condition and the large jump coefficient, by using a truncation argument. Under some extra conditions, we also derive some $L^p$ moment estimates on the solutions. As an application of moment estimates, we prove the Hölder regularity of the solutions.