We investigate the Full Counting Statistics of an electrical Mach-Zehnder interferometer penetrated by an Aharonov-Bohm flux, and in the presence of a classical fluctuating potential. Of interest is the suppression of the Aharonov-Bohm oscillations in the distribution function of the transmitted charge. For a Gaussian fluctuating field we calculate the first three cumulants. The fluctuating potential causes a modulation of the conductance leading in the third cumulant to a term cubic in voltage and to a contribution correlating modulation of current and noise. In the high voltage regime we present an approximation of the generating function.