A negative cycle only causes infinite score if you can use it on a path from to . The cycle must be both reachable from the source and able to reach the destination. Run BFS from node to find all reachable nodes. Run BFS on the reversed graph from node to find all nodes that can reach .
A node can reach in the original graph if can reach it in the reversed graph. If a negative cycle node is both reachable from and can reach , output . Otherwise, the cycle is irrelevant, and you output the negated shortest distance.