A self-consistent rate-equation (RE) approach to irreversible island growth and nucleation is presented which takes into account cluster mobility. As a first application, we consider the irreversible growth of compact islands on a two-dimensional surface in the presence of monomer deposition (with rate F) and monomer diffusion (with rate D(1)) while the mobility of an island of size s is assumed to satisfy D(s)=D(1)s(-μ) where μ>0. Results are obtained for the dependence of the island-density and island-size distribution (ISD) on the parameters D(1)/F, μ, and coverage θ. For all values of μ, we find excellent agreement between our self-consistent RE results and simulation results for the island and monomer densities, up to and even somewhat beyond the coverage corresponding to the peak island density. We also find good agreement between our self-consistent RE and simulation results for the portion of the ISD corresponding to island sizes less than the average island-size S. However, for larger island sizes the effects of correlations become important and as a result the agreement is not as good. Using our self-consistent RE approach we also demonstrate that the discrepancies between simulations and recent mean-field predictions for the exponent τ(μ) describing the power-law size dependence of the ISD for μ<1 can be explained almost entirely by geometric effects. Our results are also compared with those obtained using a simpler mean-field Smoluchowski approach. In general, we find that, except for the case μ=1/2 (for which the island and monomer densities are reasonably well predicted), such an approach leads to results which are in poor agreement with the simulations.