sounds to me a bit like young dandelions. If it is, you need to dig the whole long thick root out, if you want to get rid of it.
Having said that, I have one place in my garden where I encourage dandelions. I get lovely yellow flowers in the spring. Then I dead-head the flowers quickly, before they form the dandelion "clocks". Otherwise my garden, and my neighbours' would get over-run.
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