I don't really know which plant/weed this is, Tilly - could you please post a photo of it?

There isn't a plant called Native bur, but there are several with bur in their name. I wonder if its cleavers/sticky annie/goose grass that has the small round bur seeds. but if its in the lawn could be one of the ones that are members of the parsley group. A photo would help.

Are they rooted in the gravel/slate or are they coming up through the layers.?
The lawn will need feed and weed type treatments and in the gravel pull/hoe or weed killer it. Either way remove any flower seed heads before they disperse.

Butterbur is the common name for Petasites. Petasites albus is invasive and is not native, Petasites hybridus is the native Butterbur. Removing the flowerhead as Seaburngirl has said will help. Strimming over the plant and not letting it grow will kill it off, esp if you apply a herbicide.

but is this the plant Tilly means. there is also burdock, bur reed etc.

Or do you mean goosegrass, the long twining one that has little burrs that stick to your clothes? If that's it pull it all up before it sets seed. its only annual so the more you clear one year the less you'll get the next.