We study two-type competing first-passage percolation on random graphs generated by the configuration model with a power-law degree distribution with exponent \tau in (1,2), corresponding to the infinite-mean regime. In the classical nearest-neighbor setting, the competition is dominated by giant-degree hubs: the type that first reaches a hub rapidly infects the entire network, leading to a "winner takes it all but one" phenomenon. We extend this model by introducing long-range infections: each infected vertex infects a uniformly chosen vertex at rate \gamma>0, independently of the edge-based dynamics. This global transmission mechanism competes with the local spread and fundamentally changes the phase diagram. In cybersecurity terms, this models malware or information campaigns that spread both through local network connections and via global mechanisms, such as phishing, mass email, or broadcast exploits, which can reach arbitrary devices. This provides a natural framework for studying attacks on heterogeneous networks, many of which have heavy-tailed degree distributions with \tau in (1,2) or \tau in (2,3). We identify a sharp threshold for coexistence as a function of $\gamma$. In the subcritical regime, the "winner takes it all" phenomenon arises, with the losing type infecting either finitely many vertices or even infinitely many but a vanishing proportion of the graph. In the supercritical regime, long-range transmission enables macroscopic coexistence, including an extreme case in which the final proportions of the two types converge to a random limit characterized by a Pólya urn.