Answer: Tosafos (Shabbos 40b) teaches that while a pot on a flame (kli rishon) retains its heat even after it’s been removed from the flame, a kli sheni cannot cook raw food placed into it.
Tosafos (Shabbos 42a) writes that there is a machlokes as to whether iruy kli rishon, pouring from a kettle, is equivalent to a kli sheni, secondary container, or if it is considered to be cooking kdei klipa, the outermost food that it is poured over. R’ Ovadia Yosef (Yabia Omer OC 3:33:16) writes that there are rishonim who hold that it is the same as a kli rishon¸ while other rishonim hold that it is like a kli sheni. The Shulchan Aruch (OC 318:10) writes that one cannot pour from a kli rishon onto spices. The Mishna Berura (318:74) explains that we pasken that iruy kli rishon cooks kdei klipa.
R’ Yitzchak Meltzen (Shevisas Hashabbos, Hakdama to Bishul 19) relates that R’ Aryeh Leib of Stavisk ensured that he only poured hot water into a dry cup, though no one else adhered to this practice, and it is unnecessary. R’ Ovadia Yosef (Yabia Omer OC 3:33:1) explains that this difference in opinion is actually a machlokes as to whether pesik reisha delo nicha lei, (where one has no interest in the inevitable benefits), is muttar or assur miderabanan.
R’ Moshe Feinstein (Igros Moshe OC 1:93; 4:74 Bishul 19) writes that one mustn’t pour hot water into a cup that contains wet droplets, as one will be heating those droplets from a kli rishon. While R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 9:30) writes that one can shake out the cup so that it is relatively dry, R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 1:52; n153) and R’ Moshe Stern (Baer Moshe 6:110) write that the cup must be totally dry.
R’ Eliezer Waldenberg (Tzitz Eliezer 13:40), R’ Menashe Klein (Mishne Halachos 6:67) and R’ Shmuel Wosner (Shevet Halevi 7:42:2), however, write that that these drops are inconsequential and that there is no melacha involved by heating them up.
In conclusion, if the kli sheni contains cold droplets, it is ideal to shake out the droplets before pouring in hot water.