How much does the core of the sun wobble and does this affect light reaching the Earth?
The core of the Sun is stable to convection - the energy transport is dominated by radiative diffusion. The core of the Sun is therefore not turbulent and not wobbling around.
However, the Sun itself does "wobble" because it has characteristic oscillation frequencies that are excited by convective motions in its outer parts. The amplitude of the oscillations are about 0.2 m/s and have a period of about 5 minutes (Kjeldsen et al. 2008). This equates to amplitudes of radial motion of around 60 m, or about 1 part in 10 million compared with the solar radius.