No. You will need to remove the turf and thoroughly dig the ground, especially if it has not been cultivated before.
Blackcurrants are hungry plants and you should dig in lots of well rotted compost or well rotted manure if you can get it. If you plant into poor uncultivated soil you will get poor growth and crops. Membrane is not really suitablefor fruit and veg plots. If when the plants are established you wish to use something to deter weeds you can use a mulch, either organic or a bought one designed for going round trees.
But you can't plant fruit bushes and just leave them, they need work.