Recalculating shortest path after changing the weights
The standard term for this kind of problem is "dynamic shortest paths" or "shortest paths in a dynamic graph".  Here "dynamic" refers to the fact that the graph might be updated, and you want to update the shortest-paths information (hopefully more efficiently than re-computing it from scratch).  There are many kinds of updates to the graph one can consider: deleting a vertex, inserting a vertex, deleting an edge, inserting an edge, increasing the length of an edge, decreasing the length of an edge.  Your problem is the special case where we only have to deal with decreases to edge weights.

So, I suggest you search the research literature (e.g., using Google Scholar).  You should find a bunch of papers that discuss algorithms for this situation.