Work on clean metal. It's not a deep penetration rod and therefore does not burn though surface contaminants very well.
Having a slag stringer in the start of a 6013 bead is very common. What you are seeing is the heavy flux on top of the weld pool, and not molten metal. Getting a slightly slower start to the weld bead will solve this problem. A little hesitation before moving the rod will give the puddle time to bridge between both surfaces and solve this problem.
Establish a weld pool at start: Try pausing when you first strike up so that a puddle can form and slow the travel speed down a little for the first inch. Try doing a very small circular motion at the start.You can see when the pool has formed by the bright semi circular ring at the trailing edge of the arc.
Small 6013 rods (2.4mm or less) take a different technique. They need a real short arc and up the current to compensate.There is more slag than metal, so you need to keep the rod under the slag. They need to be dragged as the metal burns back more than the flux. Look a a used rod and you'll see the metal is recessed. Keep the angle at about 70 degrees. Too vertical will allow slag to flow ahead of the weld.
Arc length, electrode angle, current settings, welding speed, can all affect the formation of the pool,
holding too long of an arc length, and it is difficult to manipulate the metal where to go. Put the rod right into the crack
1.6 rods are difficult to learn 2mm aren't much better. 2.5 are the best place to start, you can get away with a lot, but picking up bad habits isn't easy, with 3.2mm rods you can just drag the end of the rod on the metal and it will weld, try that with a 2.5 and you will find problems. 1.6mm rods burn away faster than a sneeze and are hard to control.
1.6 and 2mm 6013's violate the “standard” for arc welding. You need to drag the rods, not maintain a gap. They will maintain their own arc length inside the flux coating. You will find on the back side of the rod, the flux will extend down to the weld pool and maintain the correct arck length. If you lift the rod, the flux will get in front of the weld and you will have problems of slag inclusions.