Hi Tom,
Great that you enjoyed the course.
On these weeds, I understand your concern and it can look daunting at this peak time of their regrowing, thanks to the mild winter and current warmth.
I felt similarly apprehensive about a few such beds last spring, but found that removing the new stems, using a trowel to extract 4-6in of any new growth, two to three times over four to six weeks, saw them expire as the parent roots had then exhausted themselves.
I would plant as normal and just keep an eye out with a trowel to hand. If you want to use polythene it is an option and helps keep potatoes white, though slugs can accumulate in the damp underneath it.