Seems its the seeds then I would weed then put weed killer down as my son does, then he only has to do it once a year.

Chick weed is pretty easy to remove, I'd just keep pulling it up rather than use weedkiller.

Agree with moongrower - unless you use a weedkiller that sits in the soil, it won't keep regrowth at bay, and those sorts of chemicals are reserved for use on paths and concrete. Usually the largest amount germinates in Spring, and the next lot are not quite so numerous. I just pull it out when I see it.

Chickweed is hardly the worst weed in the world! I'm with Moongrower & Bamboo, just pull it out - any budgies would be grateful for it!

Try pulling it out before it flowers - all your repeats are probably from letting it set seed.

Try putting a couple of inches of mulch or stone on the soil once the area is clear. Plants like chickweed prefer bare soil to germinate. Sometimes corn gluten meal will prevent weed germination, too.