For every positive integer N ≥ 2 we consider the linear differential centre in ℝ⁴ with eigenvalues ±i and ±Ni. We perturb this linear centre inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. Then if the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.