I like using cardboard and mulch around most of my fruiting bushes -- blueberries, raspberries and blackberries -- and trees -- apples, pears, peaches, etc.
It's best to keep cardboard/mulch away from the trunk because rodents can hide under and nibble on the bark, but thats a small area to maintain. I did use a few inches of sharp gravel at the base of the weeping cherry tree in the front yard because its too hard to get past the weeping branches, and that works.
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It helps to pull all the weeds one last time, so you are putting the cardboard down on bare ground. Water the ground, then lay the cardboard down and water it. Press it down a bit, so it is in good contact with the ground, then put mulch over it.
Itsa blueberry blackcurrant and raspberry. So pull weeds out place flat pieces of card over soil and cover with more soil. Will be fiddly as i have lots of bedding plants too that are flowering.
Not soil -- mulch -- blueberries in particular do well with pine wood shavings or shredded pine bark mulch since they will help to keep the soil acidified and raspberries will do well with shredded bark mulch -- or wood chippered blend but you can also use straw.
If you don't have a source for wood shavings, try feed store or farm type store that sells horse and animal bedding.
Not familiar with currants since I cant grow them here, but I would think either will work.
Bedding plants shouldn't be close to the berry bushes since they would get in the way of harvesting and also will compete with the shallow-rooted berries -- same as weeds. But if flattened corrugated cardboard is too big or awkward to manage for the area, several layers of kraft paper such as used for shipping will work well too. If you are not concerned about ink and you get newspapers, then you can use thick layers of newspapers. Smaller pieces need to be overlapped, not butted up to leave gaps.
Learning never ends because we can share what we've learned. And in sharing our collective experiences, we gain deeper understanding of what we learned.
I reckon mulching is by far the best.
Under planting always carries the risk of introducing a host plant that could carry pests to your bushes, Aphids and sawfly for example, as well as drawing nutrients and water away from the cropping plants.