A series of model calculations for the evolution of the surface layers of an accreting neutron star is carried out. The neutron star mass, radius, core temperature, and surface magnetic field strength are systematically varied, as are the accretion rate onto the neutron star surface and the metallicity of the accreting matter, in order to determine the effects of these parameters on the properties of thermonuclear flashes in the surface layers and the emitted X-ray bursts that result from such flashes. The core temperatures required for thermal equilibrium are found to be approximately a factor of 2 lower than estimated in earlier work. Owing to the effects of the gravitational redshift, the emitted X-ray bursts have lower peak luminosities and longer durations than those calculated in the Newtonian approximation. The entrainment of hydrogen into helium flashes can cause the flashes to exhibit a rather wide range of observable effects and can decrease by a factor of more than 2 the ratio of persistent accretion-driven luminosity to time-averaged burst luminosity emitted by the neutron star.