That looks like it might be Nightshade, a noxious "weed" from the same family as the Tomato, Pepper, Eggplant, that seeds readily and will grow from any bit of root left behind. These plants are quite resistant to herbicides, "weed" killers, so the best means of control is digging them out being sure to get all of the root

Here is a link that might be useful: About Nightshade

could you post a clearer picture of the leaves? To me, it looks like bindweed, right down to the white root. I've had several nightshades to deal with, and they didn't have that long white root thing going on.

Japanese Knotweed? And another request for a clearer picture.

Looks like Bindweed to me too. Calystegia sepium. You need to follow the roots as far as you can and remove them. As you have discovered they break off if you just pull so you need to dig them out. Every piece you leave will resprout. Or you could try the trick of bundling the stems together and immersing in a pot of Roundup. But I've only ever fought it via digging.

Well, Round Up had little effect two weeks ago. I just spent two days digging the buggers out. Long, long roots on those weeds. Let's see how long it takes for them to grow back.

And boy, am I sore.

Andrew

Just keep pulling up the growth that shows up (and it will, sadly), and you will eventually exhaust the mother root (or whatever it's called) and the little (*&&%^$F@ will stop showing up.