Finding a path with certain properties in a directed graph
If all partitions contain only a single vertex and you're given all of them as your initial set of $k$ vertices then the path you seek must be a Hamiltonian path. So it's NP-complete.

However, it's fixed-parameter tractable when parametrized by $t$ — it can be turned into a reachability problem on a larger graph where each vertex is a pair $(S,v)$ of a set $S$ of the partition sets whose valid point has already been passed and a vertex $v$ of the original graph. So the time is $O(\binom{t}{\le k}(m+n))=O(2^t(m+n))$.