Purslane should not be a problem in a properly maintained and established lawn but it can grow in a newly seeded or otherwise "threadbare" lawn. It's an annual so if you can get the lawn established, water and fertilize it as needed, the purslane should cease to be a problem. I would mow the lawn at the height of the tips of the grass to chop off the offending purlane tops. Keep doing that until you can start to mow the lawn proper.. As the lawn thickens up the purslane should be less noticeable and it should not get started again next year. Depending where you are located, late summer/early fall is the best time to seed a new lawn because there are fewer weed problems, less heat stress on the new grass, and warm soil for good germination and growth.
