Tuning the electronic properties of graphene by adatom deposition unavoidably introduces disorder into the system, which directly affects the single-particle excitations and electrodynamics. Using angle-resolved photoemission spectroscopy (ARPES) we trace the evolution of disorder in graphene by thallium adatom deposition and probe its effect on the electronic structure. We show that the signatures of quasiparticle scattering in the photoemission spectral function can be used to identify thallium adatoms, although charged, as efficient short-range scattering centers. Employing a self-energy model for short-range scattering, we are able to extract a d-like scattering potential delta = -3.2 +/- 1 eV. Therefore, isolated charged scattering centers do not necessarily act just as good long-range (Coulomb) scatterers but can also act as efficient short-range (delta-like) scatterers; in the case of thallium, this happens with almost equal contributions from both mechanisms.