We study theoretically the effects of pump-spot size and location on photon condensates. By exploring the inhomogeneous molecular excitation fraction, we make clear the relation between spatial equilibration, gain clamping, and thermalization in a photon condensate. This provides a simple understanding of several recent experimental results. We find that as thermalization breaks down, gain clamping is imperfect, leading to “transverse spatial hole burning” and multimode condensation. This opens the possibility of engineering the gain profile to control the condensate structure.