Magnetospheric processes at Mercury are investigated to determine how they affect the source and loss rates of the neutral He and H atmosphere. The atmospheric source rate caused by direct impact of the solar wind on the planetary surface is estimated by using a model that included the effects of the large planetary core upon the compressibility of the magnetosphere. As the nonconducting planetary surface could inhibit Birkeland currents that cause erosion of the dayside magnetosphere, the impact of the solar wind on the surface is modeled with and without the effects of erosion. The solar wind impacts the surface, at most, only about 6% of the time, and the resulting atmospheric supply rate is negligible. The flux on closed dayside field lines is estimated to be roughly comparable to the flux to the surface along open dayside field lines in the polar cap region. The flux to the surface from the plasma sheet is estimated as being about three times either of these sources, provided that all sunward convected particles in the tail precipitate to the surface. A comparison of the flux available from convection with the flux that would precipitate to the 'auroral zones' under the assumption of strong pitch angle scattering shows that the strong pitch angle scattering assumption may be invalid at Mercury, owing to the large size of the loss cone. Observations of a correlation between the lunar atmosphere He content and Kp, and consideration of surface processes, suggest that a significant fraction of the He striking the surface eventually enters the atmosphere. Loss mechanisms are also investigated. The extent to which magnetospheric convection recycles photoions (created at both high and low latitudes) back to the planetary surface, and to a lesser extent the uncertainty in the number of atoms exposed to sunlight, leads to considerable uncertainty in atmospheric loss estimates. An examination of atmospheric losses caused by electron impact ionization in the 'auroral zone' indicates that this mechanism does not compete with photoionization. The estimated solar wind supply of He to the surface is in the range 3.9¿1022-2.3¿1023 s-1, the estimated radiogenic supply is between 6.9¿1021 and 4.6¿1022 s-1, and the loss rate is estimated as between 1.2¿1022 and 1.1¿1023 s-1.