Influence of acceleration of electron on relativistic nonlinear Thomson scattering in tightly focused linearly polarized laser pulses is investigated for the first time. In the framework of classical electrodynamics, it is deduced and found that the more severe the change of the electron transverse acceleration, the stronger the asymmetry of the radiation angle distribution and the greater the transverse acceleration, the greater the radiation energy. Tightly focused, ultrashort, and high-intensity lasers lead to violent electron acceleration processes, resulting in a bifurcated radiation structure with asymmetry and higher energy. Additionally, the change of the initial phase of the laser brings about the periodic change of the acceleration, which in turn makes the radiation change periodically with the initial phase. In other cases, the radiation is in a symmetrical double-peak structure. These phenomena will help us to modulate radiation with more energy collimation.