Is the search for a k-Hamiltonian Path NP-hard?
Any problem in NP that contains an NP-complete problem as a special case is NP-complete.

Any algorithm that can solve the $k$-Hamiltonian path problem must, in particular, be able to solve the case $k=1$, which is just an ordinary Hamiltonian path.  We can obviously verify a claimed $k$-Hamiltonian path in polynomial time, so the problem remains in NP. Therefore, $k$-Hamiltonian path is NP-complete.