  The generic homomorphism problem, which asks whether an input graph $G$
admits a homomorphism into a fixed target graph $H$, has been widely studied in
the literature. In this article, we provide a fine-grained complexity
classification of the running time of the homomorphism problem with respect to
the clique-width of $G$ (denoted $\operatorname{cw}$) for virtually all choices
of $H$ under the Strong Exponential Time Hypothesis. In particular, we identify
a property of $H$ called the signature number $s(H)$ and show that for each
$H$, the homomorphism problem can be solved in time
$\mathcal{O}^*(s(H)^{\operatorname{cw}})$. Crucially, we then show that this
algorithm can be used to obtain essentially tight upper bounds. Specifically,
we provide a reduction that yields matching lower bounds for each $H$ that is
either a projective core or a graph admitting a factorization with additional
properties -- allowing us to cover all possible target graphs under
long-standing conjectures.
