  A subset of a vector space $\mathbb{F}_q^n$ is $K$-additive if it is a linear
space over the subfield $K\subseteq \mathbb{F}_q$. Let $q=p^e$, $p$ prime, and
$e>1$. Bounds on the rank and dimension of the kernel of generalised Hadamard
(GH) codes which are $\mathbb{F}_p$-additive are established. For specific
ranks and dimensions of the kernel within these bounds, $\mathbb{F}_p$-additive
GH codes are constructed. Moreover, for the case $e=2$, it is shown that the
given bounds are tight and it is possible to construct an
$\mathbb{F}_p$-additive GH code for all allowable ranks and dimensions of the
kernel between these bounds. Finally, we also prove that these codes are
self-orthogonal with respect to the trace Hermitian inner product, and generate
pure quantum codes.
