Finding path with specific length in weighted graph
Your problem is NP-complete by a reduction from Hamiltonian $st$-path, even with unweighted arcs. Thus, you should not expect a polynomial-time algorithm. I'll sketch a reduction below.

Let $G$ be an instance of Hamiltonian $st$-path with all $m$ arcs having weight 1. Let $G'$ be a graph obtained from $G$ by adding a long directed path of length $C$ starting from $t$ ending at (say) $t'$. Now, if there's an Hamiltonian path from $s$ to $t$, there is also a path from $s$ to $t'$ of length $m$ plus $C$. On the other hand, if there's a path of length $m+C$ from $s$ to $t'$ in $G'$, you can chop off the long path from $t$ to $t'$ to obtain a Hamiltonian $st$-path in $G$.