Solution? Put up a small fence to block off this strip of land. After visiting both Home Depot and Lowe's and considering various options, we decided to go with the simplest materials. We got a roll of 4' wire mesh, two 6' yard posts, and two bags of concrete. (Got one bag at first but it wasn't enough. Luckily, Lowe's was just a few minutes away.)
The posts were easier to install than I thought. I thought we'd have to dig to the depth of 2 feet but we didn't. Once the holes were deep enough to place the posts, I merely stepped on the posts and they cut through the dirt to the required depth. Added the concrete and let it set overnight. By the next day, the posts were solidly in place. Then we took about 14 feet of wire mesh and wrapped it around the posts to make a 2-layer 6-foot-wide barrier. Fastened the mesh to the posts with wire ties, painted the mesh, and that was that.
I was out for the next 3 days so I didn't see the effect until Thursday. Well, it works. No more kids cutting through. Also, since this was no longer a viable path, they didn't stray into the front yard either.