We examine the interaction between intense laser pulses and strongly magnetised plasmas in the weakly relativistic regime. An expression for the electron Lorentz factor coupling both relativistic and cyclotron motion nonlinearities is derived for static magnetic fields along the laser propagation axis. This is applied to predict modifications to the refractive index, critical density, group velocity dispersion and power threshold for relativistic self-focusing. It is found that electron quiver response is enhanced under right circularly-polarised light, decreasing the power threshold for various instabilities, while a dampening effect occurs under left circularly-polarised light, increasing the power thresholds. Derived theoretical predictions are tested by one- and three-dimensional particle-in-cell simulations.