what is this and how can I kill it
This is bindweed. It is a perennial and will continue to return unless you get it under control. It will be tricky to control because it is intertwined with your glads.
It has fleshy rhizomes (roots) that store energy which it uses to resprout unless you dig up or kill the roots, too. You cannot simply remove the top growth and expect it to die, unless you watch it like a hawk and remove each bit of new growth as soon as it appears. If you do that, you'll exhaust the stored energy in the roots eventually.
Another approach is to spray the leaves with a systemic herbicide containing the ingredient glyphosate (found in Roundup and many other products.) This will travel down and kill the roots, but it is also a "total vegetation killer" which means it will kill anything it contacts. To apply it ONLY to the bindweed, you have to pull portions of the vine clear of the glads and spray, using a shield to prevent spray from touching your glads.
Whatever you do, do not let it bloom, which will produce seeds that will then drop and spread the bindweed.
Here is our webpage on bindweed: http://extension.umd.edu/hgic/bindweed