Combining Floquet theory with the hierarchy of correlations (a method based on the formal expansion into inverse powers of the coordination number), we study the nonequilibrium dynamics of the Mott insulator state in the Fermi-Hubbard model under the influence of a harmonically oscillating electric field representing the pump laser. After deriving the associated Floquet exponents, we consider multiphoton resonances leading to the generation of higher harmonics. For weak electric field strengths, the strongest signal is obtained for the third harmonic when the driving frequency equals one-third of the Mott gap (because even harmonics are absent due to inversion symmetry). For stronger electric field strengths and lower driving frequencies, the higher-harmonic spectrum can also exhibit plateaulike structures extending over several harmonics (i.e., Floquet channels) below the resonance peak, which is consistent with previous findings.