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It looks like borage, which is an herb. The mature plants have a deep taproot. That means if you pull it up, you must get the whole root or it will regrow, as you know. You can continue to dig them out as they show; young plants will pull easier, or you can use a non-selective herbicide with glyphosate that will kill the plant down to the roots. It will also kill anything else nearby that the spray touches so be careful with application. Always read and follow instructions on the pesticide label when using.