I’ve overlayered both a raised bed and an asparagus row with horse manure the winter past and have had barely a weed in the past 6 months. Yes, the odd one but it’s like 2 weeds to extract every 10 days or post rainfall. It does the worms a lot of good too. To date I harvested 5lb of Asparagus (100 odd spears) in April on a 5 sqm bed, with the expectation of about another 5lb in May based on the number of spears harvested plus visible currently. So it doesn’t do it any harm although I have no previous data to compare it to to say whether yields have improved or not.
Obviously, less good for beds you want to plant lettuce into, but spuds will love it. You can also do the same in the winter once your winter onions and garlic are growing happily – I’ve done that and have been mostly weed free there all through spring.
I don’t exactly know the depth I used but 5 – 10cm over the asparagus I think and 3 – 5cm where the garlics and winter onions were.