The mishna says that spices cannot be put into a kli rishon even after it is removed from the fire because it will cook, but can be put into a kli sheini because they cannot get cooked in a kli sheini. There is a contradicitonin diyukim of the reisha and seifa of the mishna whether pouring from a kli rishon (iruy) is treated like a kli rishon or like a kli sheini. Tosafos tries to prove both directions and concludes that pouring from a kli rishon cooks the outer layer - כדי קליפה.
There is a gemara earlier 39a that says that one can only put into hot water, something that has already been in hot water prior to shabbos, otherwise it would be a violation of cooking. It is unclear whether that gemara is speaking about a kli rishon or a kli sheini. Tosafos explains that since the gemara allows pouring from whatever type of kli we are speaking about onto the food, it comes out that according to those who consider pouring from a kli rishon not to qualify as cooking, we can be speaking about a kli rishon. Meaning, one can pour from a kli rishon onto the food, so long as they don't soak the food in a kli rishon. However, if we consider pouring from a kli rishon to be cooking, we must be speaking there about a kli sheini. Therefore, according to Rabbeinu Tam who considers iruy of a kli rishon to be cooking, the braisa is speaking about a kli sheini and prohibits the soaking of a food that hasn't been put into a kli sheini before shabbos. Based on this Tosafos writes that one cannot cook an uncooked food in a kli sheini, because the water is hot and it looks like you are cooking. Tosafos says that to reconcile this with our mishna we are forced to say that most foods cannot be put into a hot kli sheini, but spices are an exception because the מחזי כמבשל doesn't apply to them since their function is just to give taste to the food. In addition to the מחזי כמבשל problem of a kli sheini, the M.B. 318:42 explains the Shulchan Aruch who says that bread cannot be put into a hot kli sheini because there are things that are easily cooked which can be cooked even in a kli sheini. Since we aren't such experts as to what qualifies, we are machmir for everything. The Sha'ar Hatziyun (68) points out that the exception is water and oil, which are explicit in the gemara that they don't get cooked in a kli sheini. Oil is clear from the gemara 40b and water from the gemara 42a. Obviously, spices are also an exception.
The Chayei Adam (Hil. Shabbos 20:4) has a famous opinion that if the water in the kli sheini is יד נכוית בותת, hot enough to burn one's hand, it has the ability to cook, even in a kli sheini. The M.B. (48) quotes the chayei adam without citing anyone who argues. Rav Moshe (Igros, O.C. 4,74,4) writes that it doesn't make that much of a difference for us because we are machmir for almost all foods to consider them מקלי הבישול - easily cooked, and therefore we don't put raw foods into a kli sheini even if it is yad soledes bo. However, it would make a difference for water, oil and spices that even we are lenient to put in a kli sheini. Based on the gemara 42a it would be permissible for one to add a drop of tap water to their hot water in a kli sheini and not worry about the water being cooked. However, the Chayei Adam would forbid adding water to a kli sheini that is יד נכוית בו, and therefore wouldn't allow if if the water were too hot.
The Chayei Adam refers us to his sefer Sha'arei Tzedek (perek 2,28, Binas Adam 9) where he cites the Rambam (Hil. Ma'asros 3,15) who says that if the kli sheini is יד נכוית בו, there is cooking even in a kli sheini. He also cites the the Tur regarding an אמבטי based on the gemara 42a, and explains that there is a source for this in the yerushalmi in ma'asros.
It seems to me that Rashi 42a d.h. b'kos, would be a support for the chayei adam. Rashi explains that the reason that cold water can be added to a cup, which is a kli sheini, is because it isn't that hot, implying that if it were too hot, one could not add water. But, even Rashi quotes a second approach that implies so long as it is a kli sheini, it is not a problem. Furthermore, the Mishna 42a that distinguishes between a kli rishon and kli sheini implies that spices can be put in a kli sheini even if it is רותחין, because that is the type of kli rishon we are speaking about as well. The Ritva writes explicitly that even if the kli sheini is רותח it doesn't have the ability to cook. The rationale for a kli sheini not being able to cook is as Tosafos explains 40b that it doesn't have the heat radiating from it's walls to maintain the temperature. Therefore, it seems clear that in Hilchos Shabbos we don't regard a kli sheini as being able to cook things that aren't מקלי הבישול, just that we are machmir for most foods that they are מקלי הבישול or that it looks as if one is cooking. But, water, oil and spices can be added to a kli sheini no matter how hot it is.
Regarding the Tur that the Chayei Adam cites who speaks about the prohibition to add water to an אמבטי even though it is a kli shein, because it is very hot; that is not a question at all. Tosafos 42a d.h. aval, also considers an ambati to be a kli sheini (unlike the implication of rashi that ambati is a kli rishon and sapal is a kli sheini), but clearly writes the distinction between a כוס to which cold water can be added and a אמבטי to which cold water cannot be added, is not temperature. Rather, the distinction is that an אמבטי which regularly has very hot water, looks like a kli rishon to that people will think it's a kli rishon, whereas a cup of hot water everyone knows is a kli sheini. Therefore, Tosafos also indicates clearly that tap water can be added to a cup of steaming hot water (or tea) in a kli sheini and will not constitute cooking.