Over the past several years, my vegetable garden has been overrun with hairy galinsoga weeds. How can I prevent them from coming up, and how can I eradicate them if they do come up?

Hello, so sorry for the delay in responding to your question.  

Here is some information about Hairy Galinsoga Weeds and how you might control it. Hairy Galinsoga is also known as quickweed, shaggy soldier or gallant soldier. Hairy Galinsoga Weeds are pervasive in gradens and crop fields, can be eaten by humans and livestock and it's flowers attract beneficial insects. It tends to grow very fast and can seed multiple generations within one growing season. 

Keeping an eye on your garden for this weed, it's best to hand-weed seedlings before flowering occurs making sure you remove the roots.  Because galinsoga goes to seed so rapidly, clean up areas immediately after harvest to reduce population growth. Since the seeds can only emerge from the top 0.4” (1 cm) of soil, a thorough cultivation at shallow depth can effectively deplete the surface seedbank without bringing up new seeds. In order to control Hairy Galinsoga you can remove them by hand or mechanically (with something like a hoe) in the early growth stages, place mulch in your garden or treat with herbicides.

And, then -- if you can't beat em, join them. More interesting facts--

Ways to Use Galinsoga

In the past, galinsoga has been known as a “famine” food, as while the edible parts are nutritional, they are also known to taste rather bland. This plant can be eaten raw or cooked, but there has been some debate surrounding the safety of consuming galinsoga raw, so do be sure to start small at first. Galinsoga has also been recognized as a “pot herb”, as the taste has been said to increase after the leaves and stems have been cooked, and were most commonly used in a soup called ajiaco, a Colombian potato soup. This brief culinary history may also be the reason behind galinsoga’s alternative name, potato weed.

Nutrients found within galinsoga include calcium, magnesium, potassium, and zinc. This plant has also been shown to contain sustainable amounts of vitamins A, C, and B1, B2, and B3. Galinsoga is also comparable to dark greens, such as spinach, as the wild edible contains three grams of protein. Because of how nutritional beneficial this plant is, galinsoga has also been popularly used as a flavoring in cooking. By drying the plant parts, they can be then ground down into a powder. Although the overall flavor of galinsoga has been described as bland, the flavor, when cooked, transforms into a subtle artichoke-like flavor.

Outside of its’ nutritional and culinary uses, galinsoga has been reported to have been used as a remedy for treating nettle stings and bleeding, therefore known for its astringent properties. Other ailments that galinsoga has been noted to be used include vomiting, snake bites, diabetes, and high blood pressure. According to research, galinsoga has been found to contain ACE inhibitors, which help to increase blood flow and help to regular blood pressure. Other conditions that galinsoga has been noted to help include inhibiting hyperglycemia, hypertension associated with Type 2 diabetes, in addition to containing antioxidants.

https://blogs.cornell.edu/enychp/vegetables/weed-of-the-week-hairy-galinsoga/

https://extension.wvu.edu/lawn-gardening-pests/weeds/hairy-galinsoga

https://eattheplanet.org/galinsoga-the-intrusive-astringent-edible-weed/

Thank you so much for your question.  It is my hope that the above information helps with your battle with Hair Galinsoga.

Happy gardening!