In the recent work by Yamaguchi et al. [Phys. Rev. Lett. 123, 113201 (2019)] Cd was identified as an excellent candidate for a lattice clock. Here, we carried out computations needed for further clock development and made an assessment of the higher-order corrections to the light shift of the 5s2 $^{1}{S}_{0}\ensuremath{-}5s5p$ 3P0o clock transition. We carried out calculations of the magnetic dipole and electric quadrupole polarizabilities and linear and circular hyperpolarizabilities of the 5s21S0 and 5s5p3P0o clock states at the magic wavelength and estimated uncertainties of these quantities. We also evaluated the second-order Zeeman clock transition frequency shift.