A. Yes, that does sound like spurge. Spurge is a poor competitor with other plants. It invades open spots whether you are growing gazanias or a lawn. In open areas surrounding gazanias, the best option you have is to use surface mulch. Mulches can be rock or wood. Gazanias will tolerate both so pick either one.
|Spotted spurge leaf closeup also showing white latex coming from freshly broken stem|
Actually your lawn is acting like living mulch. As long as you keep your lawn a solid covering and healthy, it will be difficult for any weed to invade.
Make sure your mowing height is high enough to shade the soil surface. Weed seeds like to see light if they are to germinate and invade. Secondly, fertilize your lawn regularly to maintain its density. Fertilizers that help to maintain its leaf density and shading of the soil will be high in nitrogen.
Thirdly, make sure your sprinkler system covers the entire lawn with water as evenly as possible. Most homeowners do not understand how to design a good sprinkler irrigation system. If you are going to design it yourself, make sure you learn how to do it correctly and don't just assume you know what you're doing.
Any weakness in the irrigation system will result in brown patches, dead spots in the lawn and invasion by weeds.
Make sure your sprinklers pop up high enough so they throw water from above the top of the grass. For heaven's sake do not use 2 inch pop-ups unless you have bermudagrass. Do not trim around your 2 inch pop-ups because they are too low or you definitely will see weed invasion in the lawn.