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Answer: There are too many permutations to cover, but we will concentrate on the principles and the main cases and give you a few references to sources dealing with related complexities.
One of the major rules of berachot on food is that regarding foods with different berachot eaten "together" (which is hard to define in a phrase) we make the beracha of the "main" food (ikar) and are exempt from making the beracha on that which is subservient (tafel) to it (Berachot 41a). This is certainly true when the two foods are ingredients of one combined food, e.g., soups (ibid. 36a). It is far from trivial to determine what the main ingredient is in vegetable soups that have a majority of water, cooked vegetables (in a variety of forms), and sometimes, some chicken or meat (see Living the Halachic Process, vol. VI, B-2,3). We will assume for the purpose of this question that the beracha on the soup without the soup nuts is Borei Pri Ha’adama (=Ha’adama). We will also assume that the soup nuts are made from one the five main grains and that the way it is prepared, its individual beracha is Borei Minei Mezonot (=Mezonot) (V’zot Haberacha p. 397).
One of the major determining factors of what the ikar is that we follow the majority of the mixture (Shulchan Aruch, Orach Chayim 208:7). However, a sub-rule of the laws of ikar and tafel is that if one of the ingredients is from the five grains in the form that its beracha is Mezonot, then even if it is significantly less than a majority of the mixture, the berachot are Mezonot with an after-beracha of Al Hamichya (Berachot 36b; Shulchan Aruch, ibid. 2). An exception to this sub-rule is when the purpose of the grain food is of an unimportant, technical nature, such as to make the other food stick together (ibid.; Mishna Berura 212:13) or to "stretch" the more expensive ingredients (e.g., bread crumbs in hamburgers). Therefore, the simple answer is that no matter what the nature of the soup is, if there are Mezonot soup nuts, then it apparently has some taste significance to the person who put them in, and the beracha on the soup is only Mezonot. This is true even if there may be spoonfuls of soup that happen not to have soup nuts in them, although one should be careful that the first spoonful, after making the beracha of the ikar, contain some of the ikar (Rama, OC 212:1 and Mishna Berura ad loc. 10).
The situation is different when the soup nuts play a very minor role in the eating, which would most commonly occur when there are only a small number of them, so that one would make Ha’adama on the soup, as it is not subservient to the soup nuts. This is all the more so regarding soup nuts (as opposed to kneidlach that are cooked in the soup, which is a more complicated question), which are put into the soup in the bowl, after the soup was cooked independently (see Dirshu 205:18). On the other hand, because the soup nuts are from the main grains, they cannot be ignored, and therefore Mezonot is also called for (Mishna Berura 205:11). In such a case, it is better to make the beracha first on the soup without eating any soup nuts right away (even though usually Mezonot has precedence) because if we were to make Mezonot first, we would have a doubt whether the Ha’adama is required for the soup (Mishna Berura 208:23). One can also get out of doubt and/or keep the regular order of berachot if he takes out each element from the bowl and eats them separately (ibid. 205:11). This is easier and more logical to do with kneidlach than with soup nuts or noodles. Regarding the beracha acharona, in a case in which neither is batel to the other, we would also make each beracha separately, if one ate a k’zayit of each within the requisite time (Mishna Berura ibid.).
It is agreed that if one had no interest in the soup nuts and some just happened to have fallen in, then they would be totally ignored (V’zot Haberacha, p. 117) and one would make just Ha’adama and Borei Nefashot on the soup.

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