Hi,
You need to get rid of them and then cover up the soil to keep them away.  There's a thing called the weed seed bank which has to do with the weed seeds in the ground just waiting for a little light and sun to germinate.  In addition, you have all the new weeds that spread by rhizomes (roots) like honeysuckle and all the new weeds that come from the weeds that flowered and then went to seed, thus adding new weeds.  
How to get rid of what you've got depends on what you've got.  Woody weeds like honeysuckle need to be cut and dug and/or treated with herbicide.  Herbaceous weeds with non-woody stems that just grew this year can be pulled, dug or killed with herbicide.  But as soon as you clear an area, you need to cover it with plants or ground cover (like your periwinkle) or cover it up with mulch.  The periwinkle should transplant easily as long as you water it in and keep it watered.
You could also kill areas by solarizing.  Here's an article about that:
https://extension.umn.edu/planting-and-growing-guides/solarization-occultation#:~:text=What%20is%20solarization%3F,pathogen%20populations%20in%20the%20soil.
Let me know if you have any other questions and good luck!
Barb