hello ‘all’, so my quad would not start until I removed this wire in the middle. So where do I go from here, now that it started and ran, it would not shut off..so I unplugged the CDI box and it shut off.
Now what?

 

That black/white wire is the kill wire, looks like it is earthed when it shouldn't be. They usually go to two kill switches, the one on the handlebars and another inside the ignition switch. Most likely one of those has gone bad, find out which and change it. Some kids quads also have extra kill switches, either a "sink plug" at the back, so an adult can run behind holding a string attached to the plug, and pull the plug if the child gets into difficulties, or a "remote" radio operated version of the same idea.

 

can you by pass this at all? Or no ?

 

Yes, just run the black/white from the CDI to any on/off switch, with the other side of the switch earthed, but that is exactly what the kill switch on the handlebars, and the one inside the ignition switch, should be doing.