Ask Mr. Smarty Plants

Okay, first step: go to a hardware or garden store and buy some leather work gloves.  Those burs will even get through that some times, but it's better than what you are doing now, which is hurting yourself. Then, when you pull a bur, get it down close to the roots and get as much out of the ground as you can. While you are at the hardware store, see if they have what we call a "sharp-toed hoe." It is a triangular blade, with a long handle so no stooping. It's good to go out when you're mad about something, and really let those weeds have it! The sharp point on the hoe will get some of the root out, and you will feel better, too. Gather up everything, including all the bur/seeds you can find, put them in a plastic bag and send them to the landfill.

The most important thing is to keep the plant from seeding out. The best way to do that is to prevent the plant from blooming before it sets seed. Unfortunately, the flower parts are not discernible with the naked eye, and they bloom from June to November, followed by the seeds which are, of course, the burs. This means that in early Spring you need to know what these plants look like when they first come up and get as much out manually as you can. You mentioned mowing, but these plants often crawl along the ground, too low for the mower to be effective.

Here is some information extracted from a previous Mr. Smarty Plants answer. The correspondent specifically asked about the use of corn gluten as a pre-emergent herbicide. Note the comment on healthy turf being the best preventive for sandbur infestation. The reason for that is that grassburs do not tolerate shade well, so a thick cover of turf grass or pasture grasses will retard growth of the bur.

"A grass itself, grassbur is a common, troublesome weed in pastures, lawns and other turf areas.

Grassbur thrives in overgrazed or otherwise struggling turf, especially nutrient-poor turf. Healthy turfgrass will typically outcompete Cenchrus. Improved turf health is the surest strategy for ridding the lawn of grassburs.

Corn gluten is a popular and effective pre-emergent herbicide useful for some annual weeds, especially crabgrass and dandelion. We have not heard of its use for control of grassbur, but it might be somewhat effective if for no other reason than it contains about 10% nitrogen, which your turfgrass will appreciate. The timing of corn gluten application is critical. To be effective, it must be put down just before weed seeds germinate."

And, finally, an article on prevention of sandburs from California.