In the one-dimensional Bose-Hubbard model with on-site and nearest neighbor interactions, agapped phase characterized by an exotic non-local order parameter emerges, the Haldane insulator.Bose-Hubbard models with cavity-mediated global range interactions display phase diagrams, whichare very similar to those with nearest neighbor repulsive interactions, but the Haldane phase remainselusive there. Here we study the one-dimensional Bose-Hubbard model with nearest-neighbor andcavity-mediated global-range interactions and scrutinize the existence of a Haldane Insulator phase.With the help of extensive quantum Monte-Carlo simulations we find that in the Bose-Hubbardmodel with only cavity-mediated global-range interactions no Haldane phase exists. For a combi-nation of both interactions, the Haldane Insulator phase shrinks rapidly with increasing strength ofthe cavity-mediated global-range interactions. Thus, in spite of the otherwise very similar behaviorthe mean-field like cavity-mediated interactions strongly suppress the non-local order favored bynearest neighbor repulsion in some regions of the phase diagram.