Modified shortest path problem
Notice that if you allow once jumping without paying the weight cost, then the shortest path is exactly what you need.

Create 2 copies of $G: G_1,G_2$. For every edge $e=(v,u)\in G$ also add an edge $(v_1,u_2)$ between the node $v$ of $G_1$ and $u$ of $G_2$. Make those edge's weight to $0$.

Now, find a shortest path between $s_1$ and $t_2$ (using Djikstra)