  We establish a gluing construction for Higgs bundles over a connected sum of
Riemann surfaces in terms of solutions to the
$\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Hitchin equations using the
linearization of a relevant elliptic operator. The construction can be used to
provide model Higgs bundles in all the $2g-3$ exceptional components of the
maximal $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundle moduli space,
which correspond to components solely consisted of Zariski dense
representations. This also allows a comparison between the invariants for
maximal Higgs bundles and the topological invariants for Anosov representations
constructed by O. Guichard and A. Wienhard.
