The effects of non-resonating hot ions on the spectrum of magnetohydrodynamic (MHD) waves and instabilities in tokamaks are studied in the limit when the width of the hot ion drift orbits is much larger than the radial scale length of the MHD perturbations. Due to the large magnetic drift velocities the hot ions cannot contribute to the MHD perturbations directly, but two main effects of the hot ions, the hot-ion density-dependent effect and the hot-ion pressure-dependent effect, influence the MHD perturbations indirectly. The physics of both effects is elucidated and it is shown that both these effects can be described in MHD approach. A new code, MISHKA-H (MISHKA including the hot-ion indirect effects) is developed as an extension of the ideal MHD code MISHKA-D [G. T. A. Huysmans et al., Phys. Plasmas 8, 4292 (2002)]. Analytical benchmarks for this code are given. Results of the MISHKA-H code on Alfvén spectrum in a shear-reversed discharges with Ion-Cyclotron Resonance Frequency (ICRF) heating are presented. Modelling of Alfvén Cascades and their transition into Toroidal Alfvén Eigenmodes in a shear reversed tokamak equilibrium is considered. The hot-ion effect on the unstable branch of the MHD spectrum is studied for the test case of an n = 1 ideal MHD internal kink mode, which is relevant to short-period sawteeth in low-density plasmas observed in Joint European Torus [JET, P. H. Rebut et al., Proceedings of the 10th International Conference, Plasma Physics and Controlled Nuclear Fusion, London (International Atomic Energy Agency, Vienna, 1985), Vol.I, p.11] experiments with high-power ICRF heating.