<p>This work examines the second harmonic generation of a Hermite-cosh-Gaussian (HChG) laser beam with circular polarisation in a magnetised plasma. In order to explain relativistic electron motion and the distinctive transverse structure of the Hermite-cosh-Gaussian beam, it is assumed that the nonlinear current density is the cause of second harmonic generation. By using the paraxial approximation, analytical formulas for the second harmonic field amplitude are derived, reflecting dependencies on plasma density and beam characteristics (mode indices, intensity, and beam width). The wiggler magnetic field enhances the phase-matching conditions to permit resonant SHG by giving the second-harmonic photons some more velocity. Furthermore, the wiggler field will contribute to the maintenance of the cyclotron frequency, which confines the plasma electrons and increases SHG efficiency. While the HChG profile delivers mode-selective gain in SHG efficiency, circular polarisation improves azimuthal symmetry in the nonlinear current. The findings demonstrate how structured beams may be used to generate high-harmonics in laser-plasma interactions in a customised manner.</p>