Given a graphon
$W$
and a finite simple graph
$H$
, with vertex set
$V(H)$
, denote by
$X_n(H, W)$
the number of copies of
$H$
in a
$W$
-random graph on
$n$
vertices. The asymptotic distribution of
$X_n(H, W)$
was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where
$H$
is a clique. In this paper, we extend this result to any fixed graph
$H$
. Towards this we introduce a notion of
$H$
-regularity of graphons and show that if the graphon
$W$
is not
$H$
-regular, then
$X_n(H, W)$
has Gaussian fluctuations with scaling
$n^{|V(H)|-\frac{1}{2}}$
. On the other hand, if
$W$
is
$H$
-regular, then the fluctuations are of order
$n^{|V(H)|-1}$
and the limiting distribution of
$X_n(H, W)$
can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from
$W$
. Our proofs use the asymptotic theory of generalised
$U$
-statistics developed by Janson and Nowicki [22]. We also investigate the structure of
$H$
-regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also
$H$
-regular graphons
$W$
for which both the Gaussian or the non-Gaussian components are degenerate, that is,
$X_n(H, W)$
has a degenerate limit even under the scaling
$n^{|V(H)|-1}$
. We give an example of this degeneracy with
$H=K_{1, 3}$
(the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.