This is not Dollarweed (Hydrocotyle); it looks like seedlings from another species, likely the Rose-of-Sharon Hibiscus above it, which is known to be invasive (only a small number of cultivars are supposedly seedless). The plants "came back" because they're this year's generation of seedlings which overwintered and germinated this spring (so, they're different plants than the ones you sprayed last year).

There is no permanent way to keep weed seeds from germinating, though maintaining a mulch layer over any bare soil helps. The mulch should be about 2 to 3 inches thick. If the Rose-of-Sharon is the source of the seed, you would need to deadhead it (remove spent flowers before seed pods ripen), which will be a bit tedious since the shrub has such a long bloom time and each flower is only open for 1-2 days before it withers and starts ripening seed. Therefore, it will probably need dead-heading once a week for 2 or more months.

Miri

The seedlings should be easy to pull up or cut down. Seedlings have so little energy reserves stored that they usually do not tolerate leaf removal one or two times before dying off. An herbicide should not be needed, but if it were, anything labeled for broadleaf weeds in a home garden should work since there is no foliage of a desirable plant immediately next to the seedlings. You wouldn't need a systemic (absorbed, root-killing) product like Roundup, although it would work; a contact product instead (one which does not kill roots) should suffice for the same reasons cutting down seedlings this young should work (the plants can't afford to lose foliage one or two times and won't have enough energy to regrow).

As long as any herbicide (if used) doesn't soak the bark of the shrubs or coat their foliage, it should be fine. The active ingredient dicamba (which is part of several different broadleaf weed killer formulations) you'd want to avoid using atop tree and shrub roots, but it's less likely an application of that would harm the Rose-of-Sharon too greatly since you wouldn't be covering the entire root area with an application. (Still, avoid dicamba if you can.)

Miri

You're welcome.