Removing the bad/waste together with some of the good/food:
The opinion of Admur in the Shulchan Aruch: It is permitted to remove from food which has waste in it, the waste together with some of the food, such as [using a spoon to] remove a fly from a cup together with some of the liquid, being that then it does not appear like one is separating at all. [From here one can learn that it is permitted to remove the waste together with some of the food, and the Borer prohibition only applies when one removes the waste alone from the food. However, some Poskim limit this allowance to only specific cases. Admur in the Siddur completely negates this allowance, as explained next.]
The opinion of Admur in the Siddur: One may not rely on the customary permission granted that if there is waste in ones food then one removes it together with some of the food etc, as doing so is questionable of containing a sin offering liability and a prohibition of Sekilah G-d forbid. [Accordingly, it is forbidden to remove the waste even together with some of the food, and the Borer prohibition applies even in such a case. Practically, we rule like the Siddur.]
It is forbidden to separate the bad even when one takes with it some of the good.
Practically, what is one to do if a fly or other waste fell into one’s food?
If a fly or hair or any other waste or unwanted food fell into one’s soup, tea, coffee, drink, plate of food, it does not help to remove the fly/waste together with the food. By a plate of food one is to eat around the area of waste, and leave the waste positioned in its place. In a drink/soup one may pour out the fly together with some of the soup.
If one will separate only some of the bad together with good and will thus still leave some of the bad with the mixture do the Borer regulations still apply?
This is Rabinically forbidden to be done in all cases that separating would involve a Chatas obligation.
 Admur 319/24: Taz 319/13 and 506/3; Beir Heiytiv 319/19; Ben Ish Chaiy Beshalach 12; Chayeh Adam 16/2; M”B 319/61; Kaf Hachaim 319/42; Even Yisarel 9/25
 The reason: Seemingly, the reason is because when one removes the bad together with the good, it is not recognizable that one has selected bad from good, and it is thus permitted.
 However from the ruling of Peri Megadim 319 M”Z 13 [brought in Q&A] it implies that this Heter to remove the bad with the good only applies in certain scenarios.
 Admur in Siddur; Kaf Hachaim ibid concludes it is best to be stringent like Admur in Siddur; Chazon Ish 53 questions ruling of Taz ibid; See Piskeiy Teshuvos 319/44
 Such as a fly or fat over gravy, one may not remove the fat or fly together with some gravy. [ibid]
 The reason: The reason that Admur retracted his ruling from the SH”A is because since ones main intent is to remove the inedible and the good that is removed is only in order to allow him to remove the fly, therefore the good is nullified to the bad and it is considered that he is removing solely the bad from the good. [Ketzos Hashulchan 125 footnote 17; See P”M 319 A”A 15; Chazon Ish 53]
 So learns Ketzos Hashulchan ibid; See P”M 319 A”A 15
 Shaar Hakolel 1/1; Likkutei Sichos 11 p. 246; Introduction of the Rebbe to the Shulchan Aruch Harav; See Divrei Nechmia 21
The reason: Whenever there is a difference in ruling between the Siddur and the Shulchan Aruch one is to follow the rulings of the Siddur. The reason for this is because the Siddur was written later than the Shulchan Aruch and hence represents the final ruling of Admur in the given subject. [ibid]
 Shabbos Kehalacha Vol. 2 p. 93
 Peri Megadim 319 M”Z 13; Aruch Hashulchan 319