Two-dimensional crystals of classical particles are very peculiar in that melting may occur in two steps, in a continuous fashion, via an intermediate hexatic fluid phase exhibiting quasi-long-range orientational order. On the other hand, three-dimensional spheres repelling each other through a fast-decaying bounded potential of generalized-exponential shape (GEM4 potential) can undergo freezing into cluster crystals, allowing for more that one particle per lattice site. We hereby study the combined effect of low spatial dimensionality and extreme potential softness, by investigating the phase behavior of the two-dimensional (2D) GEM4 system. Using a combination of density-functional theory and numerical free-energy calculations, we show that the 2D GEM4 system displays one ordinary and several cluster triangular-crystal phases, and that only the ordinary crystal first melts into a hexatic phase. Upon heating, the difference between the various cluster crystals fades away, eventually leaving a single undifferentiated cluster phase with a pressure-modulated site occupancy. ; Comment: 21 pages, 14 figures. arXiv admin note: text overlap with arXiv:1410.7610