Clearing new plot - how can I dig out tarp?!

Hello all, first time posting but have been coming here for advice for a while. I've just taken on my allotment after 2 years of waiting. It's a small plot (10m x 10x). Not sure how long it has been unused for, but it's fairly overgrown and very uneven ground.

So far, the weeds have been strimmed to the ground. I have removed most of the bigger junk and my next plan (likely for the rest of this year) is to use a mattock, shovel and fork to try and dig up and even out the ground.
I am likely to go down the route of no dig/raised beds. As much as I would like to just cover the ground for several months and then heavy mulch and start the beds on top, the ground is extremely uneven and needs levelling out.

I've come against a rather frustrating issue. Underneath around 10cm of soil is some awful tarp. You know the canvas banners you see on railings beside the road? Those!

I have no idea why it is there and why it has been buried. There is some weed fabric underneath the concrete slab paths which is fine. But this stuff is buried under 10cm or so of soil.

Does anyone have any useful ideas to help get it out? I don't want to just work the soil above it as it really limits the root growth even if I do plan on using raised beds. Digging it out is proving frustrating as in order to pull it up I need to remove the earth on top...
You can see it in one of the attached photos. And yes, that tyre was also buried under the soil.

There is also a rather tall (around 2.5m) willow tree right in the middle of the southern side (behind the IBC in the photo). Goodness only knows why it's there but I'll also be removing that.

So yes, my questions to you are:
Is levelling the ground the most sensible thing to do, and is using a mattock/azada how you'd go about it?
How should I deal with the tarp underneath the soil - any ideas about removal or is it just a case of little and often and persistence?

Thanks for your help!

What a pain, goodness knows why anyone would bury stuff like that , but on reading people’s pleas for help on here it’s common practice. If you are able & like a bit of machinery have you looked into hiring a mini digger to make short work of it? Might be worth it in the long run, if not slow & steady wins the race. Cut off the bits you expose & don’t try & pull it up. Also take care of your back, they have a tendency not to recover from that sort of injury!

Thanks for the replies. Funnily enough several plots on my end of the site have had issues with flooding for a few years. This year they sorted it by adding drainage pipes and installing a pump. Apparently my plot was particularly bad, I wonder if the plastic sheets buried everywhere had anything to so with it!

I've got some tarp to cover the site while I go at it with my mattock. Thanks for suggesting I cut sections at a time. I had tried pulling it up but as you said it will be easy to get an injury.

Let the hard work begin!

The majority of the plot is covered in willowherb. There's a lot of buttercup too and then some Dock. Nothing too hard to manage, I hope. I'm not planning on growing anything this year, just sorting the ground out and starting compost ready for next year.