Consider the following language:

$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.

($G$ is undirected).

This answer to a related question suggests that if we don't limit the path length, then the problem is in $P$.

Also, if we omit the simplicity requirement, it's easy to decide the problem.

Assuming we do care about both length and simplicity, is $L$ decidable in polynomial time?