Thank you everybody, I'm definitely going to do it I've decided. The infestation is so bad that I just don't have the time to dig it out properly. I'm going to buy some roundup from somewhere, treat the area and then cover it with the tarpaulins again until next spring. Thankfully the top growth is quite low because I covered it up before it had really started to get going this spring so I'm hoping it won't be too hard for the chemicals to get at the root systems. I'll leave it then until next spring and dig up each section as I uncover it trying to get all the roots out whilst they're dormant.

I'll see what happens with the section I've uncovered this year but seeing how quickly and densely the weeds are coming back, I may have to use weed killer on that once the potatoes are out.

Any idea how to keep grass paths down? I have bordering grass paths and they're almost knee high every time I go. I don't have a petrol powered strimmer and no other way to cut it down other than perhaps grass shears? (is that the right term, are they grass shears?) My dad thought walking on them would keep them down but it doesn't. We only go at the weekends so it isn't getting enough trampling. How do people usually keep grass paths manageable without buying/hiring petrol strimmers?