Root problems.

Good afternoon Russell,  A gurt big block of text "as you have wrote" is very difficult to read.  Would you be a sport and bung in a couple of paragraphs?

Pretty please.  Cheers,  Tony.

You could use glyphosate, but I wouldn't cover the plot after spraying. Without a cover, when the weeds start to regrow, which inevitable they will, in some places, then you can see where you need a 2nd application.
Any area that do remain weed-free, you can plant into, as long as you follow the guidelines in the label instructions.
Tree roots will be an ongoing problem, as they will continue to grow as wide as the trees canopy. I once applied glyphosate and then made raised beds because of the roots from next-door's tree, but the roots still just grew up through the bed. Good luck!

Don't be daunted- divide it up!
I'd put a post in the centre & make four paths out from it.

You talk as if you have enough ground cover sheets for the whole plot, so I'm going to assume that. (They're the sort that let air through, yes? They need to be or they'll damage the soil.)
Cover it all but in such a way that you can uncover a quarter when you like- with the joins in the cover along the paths.

Pick the least awful quarter. When the cover has killed all the top growth so that you can see the wood for the trees, clear it as much as you can by hand over a few weeks, a bit at a time when you have half an hour; each time re-covering it when you go home. When its as good as you can get it put all the manure you can find on that quarter, cover it again & go home smug.

Next spring plant potatoes in your clear quarter & gradually, as time allows, start clearing the next. Same as last time- cover when you're not there so it doesn't go backwards in between.

Next winter put a bit of lime on quarter one and all the manure you have on quarter two. Then recover them until they're needed. In spring plant potatoes in the new quarter & brassicas in last years' .
You get the idea? You'd have a four year rotation in mind for my idea but if your dividing up skills are better than mine it could be five, or whatever you prefer.

I did this with a piece of very neglected farmer's field that was as hard as concrete and had more weeds than grass. It is a very productive plot now but I still cover any beds I'm not using over winter otherwise it produces buttercups when I'm not looking. The covers allow you not to panic because one bit isn't creeping up on you from behind while you're working away on another!

Good luck whatever you decide to do.
Windy.

Good afternoon Russell,  A gurt big block of text "as you have wrote" is very difficult to read.  Would you be a sport and bung in a couple of paragraphs?

Pretty please.  Cheers,  Tony.

@Kleftiwallah:  If you find a gurt big block of text (only ten lines long) very difficult to read, as you have wrote, perhaps you could be a good sport and bung in a nap after reading the first five lines, and before you read the next five lines? Pretty please? 

We all have our different styles of writing, please be tolerant and respectful of other members different styles of posting. The forums are a place for gardeners to ask and answer questions; grammar nannies are really not necessary.

Personally, I found Russell Atterbury's original post clearly stated the problems he faces, and the opinions he was asking for. I will try to offer some gardening suggestions in reply.

@Russell Atterbury

You have already received some good advice from mumofstig and coldandwindy regarding general weed control.  I could offer other options, but the advice you received should do the trick, so I won't muddy the water with additional options. 

Regarding the tree roots, I have similar problems in my home garden, but the tree roots are coming from pine, elm, etc. in woods that are very close to the garden beds.  My suggestion in your allotment is to dig a narrow trench outside your growing beds but as close to the trees as possible. The trench should extend at least as deep as your "plow depth" in your vegetable beds, deeper if you can. When you come to tree roots, cut them off. Tools such as heavy loppers, axe or hatchet, machete, or a digging bar with a blade are good options for root cutting.  This is certainly work that is best done when the weather is cool!

If you can get extra heavy polythene sheet (not weed control fabric) and can work it down to the depth of the trench, it may help deter roots from penetrating back into your vegetable beds.  You may still need to repeat the process every few years.

Regarding water retention by your soil:  As a soil forms over a long period of time, clay tends to move lower in the soil profile. While you have a trench open in your soil, see if you can detect an increase in clay with depth (often seen within the first foot below the soil surface). I've found that I can greatly improve water retention by turning the soil over as deeply as I can to work more clay into the surface soil.  A little clay is not a bad thing, as you can't have a good loamy soil without some clay in it.

Finally, those grass clippings you mentioned would be a good mulch to place on the soil where crops are already growing. I mulch heavily with grass clippings around squash, peppers, beans, etc.  It keeps the weeds down, and keeps soil moisture in the soil. 

Good luck to you!