  We study the model $G_\alpha\cup G(n,p)$ of randomly perturbed dense graphs,
where $G_\alpha$ is any $n$-vertex graph with minimum degree at least $\alpha
n$ and $G(n,p)$ is the binomial random graph. We introduce a general approach
for studying the appearance of spanning subgraphs in this model using
absorption. This approach yields simpler proofs of several known results. We
also use it to derive the following two new results.
  For every $\alpha>0$ and $\Delta\ge 5$, and every $n$-vertex graph $F$ with
maximum degree at most $\Delta$, we show that if $p=\omega(n^{-2/(\Delta+1)})$
then $G_\alpha \cup G(n,p)$ with high probability contains a copy of $F$. The
bound used for $p$ here is lower by a $\log$-factor in comparison to the
conjectured threshold for the general appearance of such subgraphs in $G(n,p)$
alone, a typical feature of previous results concerning randomly perturbed
dense graphs.
  We also give the first example of graphs where the appearance threshold in
$G_\alpha \cup G(n,p)$ is lower than the appearance threshold in $G(n,p)$ by
substantially more than a $\log$-factor. We prove that, for every $k\geq 2$ and
$\alpha >0$, there is some $\eta>0$ for which the $k$th power of a Hamilton
cycle with high probability appears in $G_\alpha \cup G(n,p)$ when
$p=\omega(n^{-1/k-\eta})$. The appearance threshold of the $k$th power of a
Hamilton cycle in $G(n,p)$ alone is known to be $n^{-1/k}$, up to a $\log$-term
when $k=2$, and exactly for $k>2$.
