Here's my strategy, see what you think. Clear the ground of trash and sticks, etc. Trim up any small shrubs or perennials so you can see what you're doing. Take old newspapers and make a pad with 7 layers of newsprint. Starting at the outward edge, begin layering sections of pads, like you're shingling a roof. Overlap the pads so that it's not easy for soil to wash under the edges. Do not put the newspaper right up next to shrub trunks or perennial crowns. Next, mulch at the proper depth with material of choice. Next, sprinkle the Preen over the mulch, keeping it from staying on the leaves of shrubs or other plants. This helps keep perennial weeds from coming up and starving them of light. The Preen helps prevent blown in seeds from germinating. You have to reapply the Preen after 3 months, by the way. The newspaper will degrade over a single season, but by then it's prevented many weed seeds from germinating. The earthworms like it too! If you do this regularly, weeds become less and less.

Three herbicides come to mind, all liquids in their agricultural formulations. Treflan, Sonalan, and Prowl. To a lesser degree, Eptam might do the job also. Treflan is an ingredient in Preen. Not sure if the others are available for homeowners. I assume you would be applying them to a lawn. If so, read the labels carefully as Eptam has some activity on some perennial grasses. I would probably look to Treflan (Preen) first, then Sonalan, then the others. I think Preen is a granular, but you can apply it, then water it in with about 1/2" of rainfall or sprinkler irrigation within 24 hours and it will take the herbicide into the ground.

Preen has always worked out great for me in controlling weeds. It will not hurt full sized plants and will stop weed seeds from germinating. See https://theherbladyblog.wordpress.com/2017/04/01/how-i-control-weeds-in-my-stone-pathways/

You want to use a weed killer after a pre-emergent? That's fine. Just follow the bottle instructions and you're good to go, and make sure the weeds you're killing are a) sprouted, b) actively growing, and c) killed by the weed killer you're applying. Seriously, you'd be surprised. :-)