We consider the problem of finding a Hamiltonian path or a Hamiltonian cycle with precedence constraints in the form of a partial order on the vertex set. We show that the path problem is $\mathsf{NP}$-complete for graphs of pathwidth 4 while the cycle problem is $\mathsf{NP}$-complete on graphs of pathwidth 5. We complement these results by giving polynomial-time algorithms for graphs of pathwidth 3 and treewidth 2 for Hamiltonian paths as well as pathwidth 4 and treewidth 3 for Hamiltonian cycles. Furthermore, we study the complexity of the path and cycle problems on rectangular grid graphs of bounded height. For these, we show that the path and cycle problems are $\mathsf{NP}$-complete when the height of the grid is greater or equal to 7 and 9, respectively. In the variant where we look for minimum edge-weighted Hamiltonian paths and cycles, the problems are $\mathsf{NP}$-hard for heights 5 and 6, respectively.