  We prove that for fixed $r\ge k\ge 2$, every $k$-uniform hypergraph on $n$
vertices having minimum codegree at least
$(1-(\binom{r-1}{k-1}+\binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th
power of a tight Hamilton cycle. This result may be seen as a step towards a
hypergraph version of the P\'osa-Seymour conjecture.
  Moreover, we prove that the same bound on the codegree suffices for finding a
copy of every spanning hypergraph of tree-width less than $r$ which admits a
tree decomposition where every vertex is in a bounded number of bags.
