We report measurements of the in-plane thermoelectric power (TEP) for an overdoped (OD) crystal of the single layer cuprate superconductor Tl2Ba2CuO6+x (Tl2201) at several hole concentrations (p), from 300 or 400 K to below the superconducting transition temperature (T-c). For p = 0.192 and 0.220, small upturns in the TEP below 150 K are attributed to the presence of charge density waves (CDW) detected by resonant inelastic x-ray scattering studies. This suggests that measurement of the TEP could provide a simple and effective guide to the presence of a CDW. Over a certain temperature range, often strongly restricted by the CDW, the TEP is consistent with the Nordheim-Gorter rule and the T-1 and T-2 terms in the in-plane resistivity of similar crystals observed below 160 K. Two scenarios in which the T-1 scattering term is uniform or non-uniform around the Fermi surface are discussed. As found previously by others, for uniform scattering the T-1 terms give scattering rates (tau(-1)) at lower p that are somewhat larger than the Planckian value k(B)T/(h) over bar and fall to zero for heavily OD crystals. Near 160 K, tau(-1) from the T-2 terms corresponds to the Planckian value.