  We prove a central limit theorem for a certain class of functions on sparse
rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and
vertex weights. Our proof of the central limit theorem uses a perturbative form
of Stein's method and relies on a careful analysis of the local structure of
the underlying sparse inhomogeneous random graphs (as the number of vertices in
the graph tends to infinity), which may be of independent interest, as well as
a local approximation property of the function, which is satisfied for a number
of combinatorial optimisation problems. These results extend recent work by Cao
(2021) for Erd\H{o}s-R\'enyi random graphs and additional i.i.d. weights only
on the edges.
