There are a few things you can do to get the perfect poached eggs:
1. Use the freshest eggs possible. Now, I have used eggs that are a couple of weeks old and had great results, but the fresher the better. Older eggs will be more watery which is when you get all those wispy white bits in the water.
2. Strain the eggs. This might sound a bit weird but trust me, it really works. Use one of those mini fine mesh sieves over a mug and you'll be left with the thicker part of the whites, but all the thin watery part will drain into the mug (which again, gets rid of the wispy white bits in the water)
3. Add some apple cider vinegar to the water. This helps emulsify the egg whites, which basically just means everything keeps together more.
4. Water temperature. I heat the water so a few bubbles are rising to the surface but not enough so that the water is rapidly moving - as that will break your egg up.
5. Vortex, vortex, vortex. Spin that water. I use a small whisk and circle around the outside of the water fairly quickly so you have a good whirlpool in the middle. Then, very carefully, pour the egg in from a mug which is positioned just above the water. If you are making more than one egg then this isn't ideal as you can only really do one at once. If you want to cook several at once then use a spoon to lightly spin around the egg after you've dropped it in, which will help the whites to form around the yolk so you get that great shape.