Sorry Epsom Gardener, but I don't believe it is possible to "get rid" of Mares Tails. You can keep them under control but it involves spraying with Glyphosphate.
Some people seem to manage to live with them by just pulling their heads off as they appear.
No point in trying to dig them out - the roots go all the way to Australia - and if you leave a smidgen it just regrows.
If you cover with any kind of mulch (carpet, polythene etc) it just goes sideways and comes up beyond the mulch.
Sorry again. I'm sure it's not what you wanted to hear.
I think you just have to find a way to live with it.
PS - I had it growing at the side of a driveway. Salt (of the kind used for ice and snow) was successful at getting rid/keeping under control. But no use as a method for a veg bed.

__________________

From each according to his ability, to each according to his needs.