Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph

Let G be a graph obtained as the union of some n-vertex graph Hn with minimum degree δ(Hn)≥αn and a d-dimensional random geometric graph Gd(n,r). We investigate under which conditions for r the graph G will a.a.s. contain the kth power of a Hamilton cycle, for any choice of Hn. We provide asymptotically optimal conditions for r for all values of α, d and k. This has applications in the containment of other spanning structures, such as F-factors.