Shortest path when allowed to reverse an edge
This is one of a class of similar problems which can all be handled by deriving a graph $G' = (V', E')$ from the original graph $G = (V, E)$ and then using the standard algorithm on $G'$.

Consider that at any point in your search in $G'$ you need the path information you would have at a corresponding point in the search in $G$ plus the knowledge of whether or not you have already used the bonus ability. Therefore what should $V'$ be? And following on from that, what should $E'$ be?