The Gemara in Masechet Shabbat 41b states in the name of Rav Ada bar Hama that it is prohibited to add a small amount of cold water to a hot Keli Rishon after it had been emptied, because the cold water will be heated. However, it is permitted to add a lot of cold water because the water will not get hot; it will only lose its chill. The Shulhan Aruch (318:12) rules in accordance with this Gemara.
The Gemara also discusses a different case in which one pours cold water into a Keli Rishon (off the fire) that still contains hot water. For example, one wants to add more water to a tea kettle that has been removed from the Blech. Clearly, pouring a small amount is prohibited, as the cold water will be cooked by the hot water. But what is the Halacha regarding pouring a large quantity of cold water in one shot? In such a case, the cold water will cool the hot water and all the water will be warm. Abbaye rules that it is permitted. The Rif and other Rishonim rule in accordance with Abbaye. The Rambam’s position is unclear. The Kessef Mishneh grapples with this issue and concludes that the Rambam might agree to be lenient. If so, that would be the opinion of Maran, as well. Hacham Ovadia rues that one may rely on the lenient opinion. Hacham David Yosef adds, “Hamachmir Tavo Alav Beracha”-it is praiseworthy to be strict.
It is prohibited to pour a small amount of cold water into an empty Keli Rishon, but one may pour a large amount of water.
It is prohibited to pour a small amount of cold water into a Keli Rishon off the Blech containing hot water, but one may rely on the opinion that permits pouring a large amount of water in one shot, although it is better to refrain from doing so.