One model of real-life spreading processes is First Passage Percolation (also called SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with i.i.d.~heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow, because of bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power law distribution $\mathbb{P}(ξ>t)\sim t^{-α}$, with infinite mean. For any finite connected graph $G$ with a root $s$, we find the largest number of vertices $κ(G,s)$ that are infected in finite expected time, and prove that for every $k \leq κ(G,s)$, the expected time to infect $k$ vertices is at most $O(k^{1/α})$. Then, we show that adding a single edge from $s$ to a random vertex in a random tree $\mathcal{T}$ typically increases $κ(\mathcal{T},s)$ from a bounded variable to a fraction of the size of $\mathcal{T}$, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton-Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős-Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.