Your terminology was a bit unusual, so I did not understand the problem very well in my first reply.
Instead of "cut out" I think you mean "end preparation" of the pipes to be welded. Basically this means proper beveling of the joint, which is normally 37.5°, resulting in a 75° total included angle. The bottom 1/16 inch should be ground so it is perpendicular to the pipe length to form a "land."
The pipe ends should be spaced about 1/16 inch apart, so that the first "root" pass of the weld fuses well to both lands, and projects slightly below the I.D. of the pipes. All weld slag on top of this first "bead" must then be removed by grinding or gouging. Then, according to your weld procedure, you add further beads as needed to fill the groove fully, plus a bit of rounding on the top. Again, as you go, remove all slag from each bead.
Failure to remove the slag completely typically results in "elongated inclusions." Failure to leave the gap between the pipes means that the first bead bridges over the land area, not penetrating fully to the inside of the pipe. This is one example of "lack of fusion." Some other ways lack of fusion could occur are welding current too low, or electrode travel too fast.
This is only a brief synopsis. A good reference is "The Procedure Handbook of Arc Welding," by Lincoln Electric. From your description, I would guess that both the pipe preparation and fit-up, and the weld execution, were off.
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