  We go ahead with the study initiated in [3] about a heat-equation model with
non-linear perturbation driven by a space-time fractional noise. Using general
results from Hairer's theory of regularity structures, the analysis reduces to
the construction of a so-called K-rough path (above the noise), a notion we
introduce here as a compromise between regularity structures formalism and
rough paths theory. The exhibition of such a K-rough path at order three allows
us to cover the whole roughness domain that extends up to the standard
space-time white noise situation. We also provide a representation of this
abstract K-rough path in terms of Skorohod stochastic integrals.
