In this article, we consider mean field games between a dominant leader and a large group of followers, such that each follower is subject to a heterogeneous delay effect from the action of the leader, who in turn can exercise governance on the population through this influence. We assume that the delay effects are discretely distributed among the followers. Given regular enough coefficients, we describe a necessary condition for the existence of a solution for the equilibrium by a system of coupled forward-backward stochastic differential equations and stochastic partial differential equations. We provide a thorough study for the particular Linear Quadratic case. By adopting a functional approach, we obtain the time-independent sufficient condition which warrants the unique existence of the solution of the whole mean field game problem. Several numerical illustrations with different time horizons and populations are demonstrated.