And you can do yourself a favor by getting a shovel out right now. Those woody vines tend to have roots that run all over the place and if you can pull out a five foot section of it, that's more you won't need to worry about and it's also rewarding in a personal way because you're destroying your enemy! I have a similar plant here that comes in from my neighbor's yard and it pops up randomly. This year I just ripped out the roots wherever they lead and it turned out they'd made a network all over the place. Your landscaper was an idiot for telling you additional dirt would smother it. The only kind of weed that you kill with plastic or dirt like that are things like dandelions - superficial weeds. Woody plants are entirely different. But because they store nutrients in their roots for the winter, applying Roundup or whatever now is most effective.

I'm not in your area and most of my advice won't do you a lot of good this season, but ... set aside a place to make your own compost and you don't need to buy! I get a cheap, unwanted spring hay or straw and it makes a rich compost. I add leaves, horse manure from my pasture, wood ashes and charcoal, and many other things. I double dig my beds and mix it well in, checking PH. If I must buy, I purchase cheap top soil and composted manures to mix my own. I may also add in some good potting soil so it's a three way blend but you are right, that stuff is way expensive if you need a lot of it like I do so I don't buy much of it. I also add in some of my native red clay for minerals but break it up and mix well. For mulch, I get /dump truck/ loads delivered for free from the electric company. Now mind you it's chipper shredded mixed woods, not pine bark, etc but it works great for me. I sort out larger pieces if I am using it containers around my plants and the larger bits go into the big garden or the compost pile. Some tree trimmers will also give it away for free if you call around. As noted already, there are free compost areas so check around. Probably your local Master Gardeners and any Victory Garden goers in your area will also have sources to share.

From the picture it looks like nutsedge. If there aren't too many and they're small, you might get away with pulling them up... over and over and over until they stop coming. Otherwise, your best bet is to use an herbicide called SedgeHammer. You can find it online, and you'll also need to mix it with a spreader sticker and spray with a pump sprayer. Follow the directions carefully and you should eliminate or greatly reduce the number of these in your grass... but eventually, they'll come back and you'll need to do it again, though in my case, there weren't near as many that came back the next time.

lawn warriors.. would suggest that if you properly grew your lawn. to the thick mat it is supposed to be .... then your weed problems would be a thing of the past ... and therein lies your solution ... if you can budget it.. hire pro's to bring your grass to life ... thru an annual program ... this is the easiest way for a newb... or.. DIY .... which can be a PIA... if you have no experience in the warrior cult.. lol ... i am sure there is a lawn forum .... as noted above... seeding lawn is very specific as to when it can and should be done.. based on your location ... its getting pretty late.. for new sown babe plants.. to put down enough roots.. to make it thru the heat of the summer .. as i presume you dont have an inground lawn irrigation system ... [or your lawn would be more lush]... just killing this stuff.. and leaving a patch of bare earth .... is just going to be a recipe to the next batch of summer weeds .... ken