A subset of a vector space Fnq is K-additive if it is a linear space over the subfield K C Fq. Let q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are Fp-additive are established. For specific ranks and dimensions of the kernel within these bounds, Fp-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 Fp-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are selforthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.