We consider the properties of a hyperaccretion model for gamma-ray bursts (GRBs) at the late time when the mass supply rate is expected to decrease with time. We point out that the region in the vicinity of the accretor and the accretor itself can play an important role in determining the rate and time behavior of the accretion and ultimately the energy output. Motivated by numerical simulations and theoretical results, we conjecture that the energy release can be repeatedly stopped and then restarted by the magnetic flux accumulated around the accretor. We propose that the episode or episodes when the accretion resumes correspond to X-ray flares discovered recently in a number of GRBs.