Manure and cover or cover and manure?

I've been gardening for years but am new to allotments. I'm in the process of clearing it of mostly couch grass. I got it deep rotavated as i couldn't even get a fork into it. So now digging over it and getting what roots I can out. I've done about half and my plan is to cover the rest with weed suppressing sheeting and then uncover it as i go. Am also thinking of getting a delivery of manure. Should I just cover the undig bit with the manure and then cover it all with the weed suppressing sheeting or cover it and leave the manuring until next autumn?
Any advice gratefully received!!

and then uncover it as i go.

If you have got Couch and Weed suppressing membrane you might want to plant-through the membrane in the first year - that will make it harder for the Couch to come back again.

If you will have time enough to keep hoeing / pulling out any Couch that comes back then you are probably OK without the Weed suppressing membrane - although it will help retain moisture - but might harbour more slugs too.

Frost (if we have much winter less) will be beneficial to the soil.  Putting a layer of muck on top will keep the frost out (and will insulate the soil to keep the cold in for longer when Spring arrives)

If you are going to incorporate it into the soil that's fine.

If you just want to put a layer on for a mulch (which will keep moisture in during the summer, and the worms can do some of the work of incorporating it) then I would wait until after the soil has warmed up in the spring