A 01: First, compare your local maps with the HydroSHEDS river network. If there are only a few differences, you can depict the respective river lines from the local map and burn them into the hydrologically conditioned DEM of HydroSHEDS. Then run a sink-fill process and derive a new drainage direction map and river network from the corrected and conditioned DEM. If the local river map shows generally better quality at many locations, or if the conditioning process of HydroSHEDS introduced an error that is not apparent in the raw SRTM data, you can use the local river map and burn it into the void-filled (un-conditioned) DEM of HydroSHEDS. However, you will then have to apply your own individual conditioning processes (at a minimum you will have to apply a sink-fill procedure) in order to produce a new hydrologically conditioned DEM.
Use the hydrologically conditioned DEM of HydroSHEDS. Reclassify the no-data cell that represents the incorrect sink with an appropriate elevation value. [There are different ways to achieve this, for example: make a new point theme; place a point at the location of the no-data cell and enter the desired elevation value into the attribute table; convert the point theme to a grid and merge it into the DEM.] Then run a sink-fill process and derive a new drainage direction map and river network from the corrected and conditioned DEM.