The Fröhlich Hamiltonian for large polarons in a crystal lattice is well-known. In its original derivation, Fröhlich considered a lattice of crystal ions with linear restoring forces: the phonons of the lattice are harmonic and do not interact. This approximation is valid in a wide range of materials. However, it has recently been shown that in solid hydrogen and hydrogen-rich materials at extreme pressures, the non-linear terms in the restoring force and the anharmonicity of the phonons are not negligible. Furthermore, a correct description of electron-phonon interaction in these materials is very important, due to the measurement of conventional superconductivity at temperatures above 200K in these materials. Based on earlier work by Kussow [1], we have derived three additional terms in the polaron Hamiltonian that describe 3-phonon processes, 2-phonon-1-electron processes, and 1-phonon-2-electron processes. We show that all of these terms are zero in crystals with an inversion centre. Using perturbation theory, we show that these additional terms can lead to a significant increase in the binding energy and effective mass of one polaron in the weak-coupling limit. [1] A.-G. Kussow, Large polaron in an anharmonic crystal lattice. Int. J. of Mod. Phys. B, 23, 1, 19-38 (2009)