Pigweed is fairly easy to pull out the whole root. If it Physalis peruviana, it should also pull out completely. More likely, I think it is one our native Physalis, like P. virginiana. It roots deeply like milkweed and creeps all over underground like milkweed, too. When you physically pull the above ground portion, you usually get a single vertical root with very little side roots, that breaks before it joins with the horizontal root that it attaches to deep underground. This is a perennial native that comes from the same network of roots every year.
I wouldn't bother trying to kill it chemically among other desirable plants. You wouldn't be successful. Just pull it up whenever you see it. It won't take over your garden unless you let it.
