Removing a fly or other waste from ones soup or cup of juice:
Note: The explanation behind the differences between the rulings of the Shulchan Aruch and the Siddur with regards to pouring out in-edibles have already been discussed in Halacha 7, see there. As well the difference of rulings with regards to if one may remove some of the bad with the good was already discussed in Halacha 1b. See there.
A. The opinion of Admur in the Shulchan Aruch: 
If a fly fell into a cup one may not remove only the fly from the cup being that doing so is equivalent to separating waste from food which is forbidden to be done even in order to eat right away. Rather one is to remove also some of the liquid together with the fly being that when one does so it does not appear like one is separating at all, as explained above (Halacha 14 [in the Michaber, and Halacha 18 above in the Alter Rebbe]).
B. The opinion of Admur in the Siddur:
Introduction: The following are a number of warnings and statements to remove stumbling blocks and common and frequent inadvertent [acts done on Shabbos] which according to many of the greatest of the Rishonim doing so involves a prohibition of Kares and Sekilah when done intentionally and a sin offering for when done unintentionally, may G-d atone for us.
Removing a fly from ones soup: One may not rely on the customary permission granted that if a fly or other waste falls into ones cup or plate then one removes it together with a spoon and takes out with it some liquid etc, as doing so is questionable of containing a sin offering liability and a prohibition of Sekilah G-d forbid.
Thus the only solution that remains is to pour out from the cup until the waste comes out from it.
Blowing the fly: One may not blow on [the waste] with his mouth until it is blown out, although he may blow it to bring it nearer to the wall of the cup and then tilt it and pour from it until the waste comes out. [See Q&A]
The reason that tilting it out is allowed: As since the removal of the waste is being done through him holding the cup of liquid in his hand and tilting it with his hand this is considered separating food from waste which is permitted to be done in order to drink it immediately.
Removing fat from a dish: The same applies if [the fly] fell into a plate with gravy, and even with fat that floats on the surface of the gravy, one may not throw it out through a spoon taking with it some of the liquid as this is similar to removing waste from food which one is liable on.
Summary-Removing a fly and the like from ones food:
One may blow the fly/insect/dirt to the side of the cup and then pour it out, by tilting the cup, together with some of the liquid. This however may only be done in order to drink its content right away. It is forbidden to remove the fly if he plans to eat the food later on.
One may not remove the fly with a spoon even if he takes some of the liquid out together with it. As well he may not blow it out.
May one remove pieces of cork from ones wine on Shabbos?
See Q&A in the filtering chapter.
May one move the fly or other waste to the side of the cup using ones hand or spoon?
No. This may only be done through blowing on it, as moving it with a spoon or finger involves the Muktzah prohibition.
May one blow the waste to the side even if it is covering the surface of the food to the point that it is impossible to drink it without blowing it to the side? 
No. The permission to blow the waste to the side only applies when one would in any event be able to manage to drink the liquid even without blowing it to the side. However if this is not the case then it is forbidden to blow it to the side at all, and rather one may only pour it out as it is together with some of the liquid in order to drink right away.
May one splash away the dirt that floats on the surface of a Mikvah?
One may do so even with his hands in order to allow himself to immerse in a clear area of the water. However one should take care to only splash the dirt away to the sides together with a lot of water.
May one remove fruits etc that have fallen into pebbles and dirt, from the floor?
One whose fruits have [fallen] into pebbles and dust, may be gathered one at a time in order to eat [right away]. However he may not place them into a basket or box.
The reason for this is so he not do a mundane act [which resembles Borer].
 And thus one may do so even with intent to eat later on
 Shabbos Kehalacha Vol 2 page 243 note 288. There he explains that the Alter Rebbe himself never wrote this “Halacha 14” and it was rather written by the printers. In the new Shulchan Aruch of Admur they omitted this all together.
 Doing so is prohibited due to the Melacha of Zoreh [winnowing] and not Borer.
 Based on the Siddur.
However according to the Shulchan Aruch Halacha 24 it is permitted to remove through using a spoon and taking out a bit of liquid together with it.
 Shabbos Kehalacha Vol. 2 p. 240
 So infers the Ketzos Hashulchan 125 footnote 18 from the Alter Rebbe above in the Siddur.
 Shabbos Kehalacha Vol. 2 p. 240
 So rules Peri Megadim
 Shabbos Kehalacha Vol. 2 p. 242
 So rules Ketzos Hashulchan 146 footnote 35
 Shabbos Kehalacha 14/45
 Ketzos Hashulchan 146 p. 98 number 2