Can you please tell me what this weed is; I presume it's a weed. It has appeared in large numbers in my beds. I planted wild flower seeds last year but I'm not sure if it's one of those seeds. I would be glad of any help. Thank you.
It looks like the dreaded creeping Charlie plant. Here is info on how to control it:
https://www.gardeningknowhow.com/plant-problems/weeds/kill-creeping-charlie.htm
I’ve tried pulling it out but afraid to use any chemicals in case they’d kill the plantings I want to grow.
You can use a dicamba herbicide on the lawn and it won't kill the grass. Be sure to read and follow instructions. For garden beds, hand pulling is best. Here is more:
https://www.gardeningknowhow.com/plant-problems/weeds/kill-creeping-charlie.htm
I planted periwinkle vine as ground cover, which was well-established after 3 years. When my back was turned, creeping Charlie infiltrated the bed. it’s impossible to pull it out without also pulling the periwinkle. is there any way to get rid of the creeping Charlie without having to dig up the whole bed and start over?
Unfortunately, since it's in your flower bed, you can't use herbicides on it. (Unless you want to do the painstaking task of "painting" a nonselective herbicide that contains glyphosate on the creeping charlie.) Pulling it out when the soil is wet is your best option, but of course, some will come back, and you'll have to keep an eye on it.
https://www.gardeningknowhow.com/plant-problems/weeds/kill-creeping-charlie.htm