Minimum spanning tree such that one edge can be minimised

Question

During a computer coding exam, I have encountered such a problem.

Given a list of vertexes and edges between the vertexes,and a positive number, D, what is the minimum spanning tree between the vertexes such that the sum of the lengths of the edges is minimal, provided that you may subtract any one edge of your choice in the tree by the positive number D and reduce the weight of that edge to max(0, weight-D)?

I know that Kruskal's and Prim's algorithm can all be used to solve the first part of the problem, the minimum spanning tree, and I can do it. However, I have been struggling hard for the second part, as I don't know how to minimise the weight of the tree given the number D.