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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 $\alphan$ and $G(n,p)$ is the binomial random graph. We introduce a general approachfor studying the appearance of spanning subgraphs in this model usingabsorption. This approach yields simpler proofs of several known results. Wealso use it to derive the following two new results.  For every $\alpha>0$ and $\Delta\ge 5$, and every $n$-vertex graph $F$ withmaximum 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 $F$. Thebound used for $p$ here is lower by a $\log$-factor in comparison to theconjectured threshold for the general appearance of such subgraphs in $G(n,p)$alone, a typical feature of previous results concerning randomly perturbeddense 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 $G(n,p)$ bysubstantially 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 Hamiltoncycle 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 aHamilton cycle in $G(n,p)$ alone is known to be $n^{-1/k}$, up to a $\log$-termwhen $k=2$, and exactly for $k>2$.

Embedding spanning bounded degree graphs in randomly perturbed graphs