Question: Is one allowed to wash one’s hands in the morning in the sink if there are dishes there?
Answer: The Shulchan Aruch (OC 4:9) writes that when one washes one’s hands in the morning (negel vasser), one must not reuse the water. One should not pour the water onto the floor where people may walk. Therefore, the Mishna Berura (4:20) writes that one must not give this water to one’s animals to drink. The Shaarei Teshuva (4:8) adds that one should not daven in the presence of this water either.
The Piskei Teshuvos (4:14) writes that one must be careful not to wash one’s hands in a sink where there are dishes. One who did so, would have to ensure that they wash their dishes properly afterwards.
However, R’ Shlomo Zalman Auerbach (Halichos Shlomo, Tefilla 20:7) maintains that one may wash one’s hands in a sink with dishes. As the dishes are going to be washed, there is no concern. The Mishna Berura (4:14) writes that if one touched food with unwashed hands, they should wash the food off three times, Nonetheless, R’ Chaim Kanievsky (quoted in Ishei Yisrael 20:n72) writes that when washing over dishes, they do need to be rinsed three times.
In conclusion, one may wash one’s hands in the morning in a sink, even if there are dishes there.