Topological insulators are a novel materials platform with high applications potential in fields ranging from spintronics to quantum computation. In the ongoing scientific effort to demonstrate controlled manipulation of their electronic structure by external means, i.e., the provision of knobs with which to tune properties, stoichiometric variation and surface decoration are two effective approaches that have been followed. In angleresolved photoelectron spectroscopy (ARPES) experiments, both approaches are seen to lead to electronic band-structure changes. Most importantly, such approaches result in variations of the energy position of bulk and surface-related features and the creation of two-dimensional electron gases. The data presented here demonstrate that a third manipulation handle is accessible by utilizing the amount of super-band-gap light a topological insulator surface has been exposed to under typical ARPES experimental conditions. Our results show that this third knob acts on an equal footing with stoichiometry and surface decoration as a modifier of the electronic band structure, and that it is in continuous and direct competition with the latter. The data clearly point towards surface photovoltage and photoinduced desorption as the physical phenomena behind modifications of the electronic band structure under exposure to high-flux photons. We show that the interplay of these phenomena can minimize and even eliminate the adsorbate-related surface band bending on typical binary, ternary, and quaternary Bi-based topological insulators. Including the influence of the sample temperature, these data set up a detailed framework for the external control of the electronic band structure in topological insulator compounds in an ARPES setting. Four external knobs are available: bulk stoichiometry, surface decoration, temperature, and photon exposure. These knobs can be used in conjunction to fine tune the band energies near the surface and consequently influence the topological properties of the relevant electronic states.