Hello Jeanine,

That looks like a variety of stonecrop (sedum).  Its waxy, water resistant leaves make it difficult to get herbicides to soak into the tissue.  Many broadleaf herbicides such as those that contain combinations of 2,4-D, dicamba, MCPP, MCPA or triclopyr may be effective.  A surfactant would help the herbicide infiltrate the plant, or walking on the plant and crushing it before herbicide application would help as well.

If you use any herbicide, carefully read the label for pet safe application which generally at least involves allowing the product to dry before your cat walks in the area of application.  But the manufacturer may require other precautions for pet safe application as well.

While this article addresses a different sedum variety, the treatment would be the same: https://www.canr.msu.edu/news/scouting_for_weeds_mossy_stonecrop 

If you do not wish to use chemicals, you can physically remove the sedum or you can smother it by putting plastic or cardboard over the sedum and letting the heat of the sun kill off the roots.  Then you can re-seed the area.

This article describes smothering or solarization methods:  https://extension.umn.edu/planting-and-growing-guides/solarization-occultation

Hope this helps.