  We study the regularity of local minimisers of a prototypical
free-discontinuity problem involving both a manifold-valued constraint on the
maps (which are defined on a bounded domain $\Omega \subset \R^2$) and a
variable-exponent growth in the energy functional. To this purpose, we first
extend to this setting the Sobolev approximation result for special function of
bounded variation with small jump set originally proved by Conti, Focardi, and
Iurlano \cite{CFI-ARMA, CFI-AIHP} for special functions of bounded deformation.
Secondly, we use this extension to prove regularity of local minimisers.
