It looks like Wild Violet to me. Have you ever seen it flower?
If so it is apparently difficult to take care of.
If you want to go the chemical way you can control the Wild violet in the grassy area using a broadleaf pre- and post-emergent herbicide from Home Depot (or better yet an agriculture centre). This will only touch the weed and not the grass.
I recommend against chemical in the garden area as you likely want to grow other perennials (broadleafs). Instead I would be persistent and cultivate it regularly and do not let it flower.
Even if it is not Wild Violet though you could try this if you have not already as the plant is a broadleaf and can be targeted within the grass.
Best of luck. If I am wrong I blame the effects of that interesting plant in post #10.
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"And whether or not it is clear to you, no doubt the universe is unfolding as it should."
Max Ehrmann
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