Help with blackened veg

I started planting veg during lockdown and since then have been slowly expanding my veg growing. This year I built some terraced raised beds as my garden has a steep slope and I built them in a part of the garden that was not used as it goes under  the branches of a very large lime tree  (the tree is in  my neighbours garden and is subject to a TPO so I can’t easily cut it back). The lower part of the raised beds in directly under the over hanging branches of the lime tree. I thought I would see how things would grow but as you can  see from the pictures, things haven’t gone so well. There is a black residue on many of the plants and they are not that healthy. The top of the new sleepers put in just in May are also blackened. 

I have read about sooty mold but I don’t know if this is it from the tree or something else.

So any ideas or advice? Is the veg still edible? Should I just abandon this part of the garden for veg growing and just grow something else?

any help much appreciated.

Paul