High powers of Hamiltonian cycles in randomly augmented graphs

We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers (Formula presented.), and (Formula presented.), and for any (Formula presented.) we show that adding (Formula presented.) random edges to an (Formula presented.) -vertex graph (Formula presented.) with minimum degree at least (Formula presented.) yields, with probability close to one, the existence of the (Formula presented.) -th power of a Hamiltonian cycle. In particular, for (Formula presented.) and (Formula presented.) this implies that adding (Formula presented.) random edges to such a graph (Formula presented.) already ensures the (Formula presented.) -st power of a Hamiltonian cycle (proved independently by Nenadov and Trujić). In this instance and for several other choices of (Formula presented.), and (Formula presented.) we can show that our result is asymptotically optimal.