How many random edges make an almost-Dirac graph Hamiltonian?

We study Hamiltonicity in the union of an $n$-vertex graph $H$ with high minimum degree and a binomial random graph on the same vertex set. In particular, we consider the case when $H$ has minimum degree close to $n/2$. We determine the perturbed threshold for Hamiltonicity in this setting. To be precise, let $\eta:= n/2-\delta(H)$. For $\eta=\omega(1)$, we show that it suffices to add $\Theta(\eta)$ random edges to $H$ to a.a.s. obtain a Hamiltonian graph; for $\eta=\Theta(1)$, we show that $\omega(1)$ edges suffice. In fact, when $\eta=o(n)$ and $\eta=\omega(1)$, we show that $(8+o(1))\eta$ random edges suffice, which is best possible up to the error term. This determines the sharp perturbed threshold for Hamiltonicity in this range of degrees. We also obtain analogous results for perfect matchings, showing that, in this range of degrees, the sharp perturbed thresholds for Hamiltonicity and for perfect matchings differ by a factor of $2$.