  For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow}
H$ if $G$ has the property that every \emph{proper} colouring of its edges
yields a \emph{rainbow} copy of $H$.
  We study the thresholds for such so-called \emph{anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form $G \cup
\mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least
$d >0$, and $d$ is independent of $n$.
  In a companion article, we proved that the threshold for the property
  $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_\ell$ is
$n^{-1/m_2(K_{\left\lceil \ell/2 \right\rceil})}$, whenever $\ell
  \geq 9$. For smaller $\ell$, the thresholds behave more erratically, and for
$4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned
aesthetic form capturing the thresholds for \emph{large} cliques.
  In particular, we show that the thresholds for $\ell \in \{4, 5, 7\}$ are
$n^{-5/4}$, $n^{-1}$, and $n^{-7/15}$, respectively. For $\ell \in \{6, 8\}$ we
determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are
$n^{-(2/3 + o(1))}$ and $n^{-(2/5 + o(1))}$, respectively. For $\ell = 3$, the
threshold is $n^{-2}$; this follows from a more general result about odd cycles
in our companion paper.
