What can I do about the numerous offshoots around our sycamore trees? Any ideas please?

I suspect the offshoots are reallly seedlings from the Sycamore.. You could probably kill trhem by cutting the root an inch or two below the surface. I did not think Sycamore sent out suckers. Posibly someone else can check on that?

Agreeing with Poaannua, I have found suckers appearing on roots that have been damaged i.e. with a mower. Cut back as far as you can and rub out any buds forming. Top dress the root area and leave it to nature.