Dijkstra's algorithm to compute shortest paths using k edges?
If the graph has no negative edges, the problem can be solved in $O(k \cdot (|V|+|E|) \lg |E|)$ time using Dijkstra's algorithm combined with a product construction.  We construct a new graph $G'=(V',E')$ with vertex set $V' = V \times \{0,1,2,\dots,k\}$ and edge set

$$E' = \{((v,i), (w,i+1)) : (v, w) \in E\}.$$

In other words, for each edge $v \to w$ in $G$, we have edge $(v,i) \to (w,i+1)$ for all $i$ in $G'$.

Now use Dijkstra's algorithm to find the shortest path in $G'$ from $(s,0)$ to a vertex of the form $(t,i)$ where $i \le k$.  This will be the shortest path in $G$ that uses at most $k$ edges.