  Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some
fixed $d > 0$, the distribution $G \cup \mathbb{G}(n,p)$ over the supergraphs
of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the
distribution of edge-coloured graphs arising from assigning each edge of the
random perturbation $G \cup \mathbb{G}(n,p)$ a colour, chosen independently and
uniformly at random from a set of colours of size $r := r(n)$. We prove that
such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles
whenever the edge-density of the random perturbation satisfies $p := p(n) \geq
C/n$, for some fixed $C > 0$, and $r = (1 + o(1))n$. The number of colours used
is clearly asymptotically best possible. In particular, this improves upon a
recent result of Anastos and Frieze (2019) in this regard. As an intermediate
result, which may be of independent interest, we prove that randomly
edge-coloured sparse pseudo-random graphs a.a.s. admit an almost spanning
rainbow path.
