Biomolecular clocks produce sustained oscillations in mRNA/protein copy numbers that are subject to inherent copy-number fluctuations with important implications for proper cellular timekeeping. These random fluctuations embedded within periodic variations in copy numbers make the quantification of noise particularly challenging in stochastic gene oscillatory systems. Motivated by diurnal cycles driving circadian clocks, we investigate the noise properties in the well-known Goodwin oscillator in the presence and absence of a periodic driving signal. We use two approaches to compute the noise: (i) solving the moment dynamics derived from the linear noise approximation (LNA) assuming fluctuations are small relative to the mean and (ii) analyzing trajectories obtained numerically from stochastic simulations algorithm. Our results demonstrate that the LNA can predict the noise behavior quite accurately when the system shows damped oscillations or in the presence of external periodic forcing. However, the LNA could be misleading in the case of sustained oscillations without an external signal due to the propagation of large noise. Finally, we study the effect of random bursting of gene products on the clock stochastic dynamics. Our analysis reveals that the burst of mRNAs enhances the noise in the copy number regardless of the presence of external forcing, although the extent of fluctuations becomes less due to the forcing.