Violet (Viola).

Not necessarily invasive. Could be just a very happy native species enjoying the conditions.

If it's everywhere it is not well behaved. Many violets produce cleistogamous buds which spew loads of seeds identical to the parent plant in addition to producing normal flowers that are open pollinated and produce seed.
I've been digging many five gallon buckets of Viola sororia var. priceana and still have more to dig up this year after several years of work., I don't want them in my flower beds or lawn.
I grow other species of Violas that are not problematic. The choice Viola pedata, for instance, does not produce cleistogamous buds.

Viola sororia ranks as one of my top weeds.

Everyone's entitled to their opinion about what they want to allow to grow in their own yard, and it's probably as good as fact that this particular violet is at the top of the list for being able to spread quite exuberantly in its' preferred conditions, but I would like to see unwanted natives get credit for being more than weeds (yep, another opinion.) This one makes golf balls roll inconsistently, and isn't usually too terribly bothered by weed'n'feed, hence its' reputation as a 'tough weed that invades turf' when in the grand scheme of things, the grass has invaded the native habitat of the violet.

For those who don't dislike this plant, Violets make a great ground cover under shrubs, a great light-traffic path through a wildish cottage garden. I cultivated them purposely when I lived in OH, especially around unmulched trees so the mower or trimmer didn't need to get right up to the trunk, possibly damaging it. If any show up in my yard here, they will be treated like welcome guests.

Just as easy to mow as grass in a lawn.

The sparse grass in the pic makes me wonder if that area may be too shady for grass to flourish. If so, and you don't want violets, something besides grass may be in order there. Is this under a tree? If so, you could cover the area with the trees' leaves when they fall (if deciduous,) that should be enough to finish killing the grass, and provide a shady bed to plant in. Leaves will not kill the violets though. If you don't want those, digging up the roots is necessary to evict them.