Find the lightest weight of a path between $u$ to $v$ that contains no more then $k$ edges (If there's one)
Consider the dynamic programming formulation of Bellman-Ford:

$\qquad \displaystyle \mathrm{bf}(i,j) = \begin{cases}
                             0         &amp;, i = 0 \land j = s \\
                             \infty    &amp;, i = 0 \land j \neq s \\
                             \min\limits_{k \in V}\, \big(\, \mathrm{bf}(i-1,k) + c(k,j)\,\big) &amp;, \text{else}
                          \end{cases}$

What meaning does $\mathrm{bf}(i,j)$ have, that is what does DP-matrix entry $\mathrm{bf}[i,j]$ contain?


   If you unfold the definition (i.e. proof by induction) you see that $\mathrm{bf}(i,j)$ is the weight of the shortest path from starting node $s$ to $j$ with at most $i$ edges -- exactly what you are looking for.