Remove the existing grout by rubbing a grout saw back and forth in the tile joints. The saw has carbide ridges that will remove the grout. Repeat with each tile that you want to regrout.
Wipe down the counter with a damp sponge to remove any grout dust. Fill one bucket with clean water. Mix the grout in the other bucket by pouring in the sanded grout, then adding water as you stir until the grout has the consistency of cake icing.
Let the mixed grout sit for three minutes. Scoop grout onto the tiles. Spread it into the joints with a wet rubber grout float. Push the grout at a diagonal to the tiles while holding it at a 45 degree angle, making as many passes as necessary until all the grout is in the joints.
Dip the sponge in the bucket of clean water. Wring out the sponge. Wipe to tiles in a circular motion to remove the excess grout. Rinse the sponge in the bucket and repeat the wiping until the tiles are clean.
Repeat the spreading of the grout with the float, and the cleaning of the tiles with the sponge, until you have regrouted all the sink tiles. Wipe the tiles with a paper towel to remove the haze, and leave the grout to dry overnight.