sickly lemon tree

Looks like Sooty mould - but I'm no expert

https://www.rhs.org.uk/advice/profile?PID=770

Looks like you've got scale insect on there. I'm no expert though this could be separate to the black.

Scale insects produce sticky gunk that goes black and is known as sooty mould in small amounts the mould is not a problem and will go when the scale insects are gone
you need to squish any of the scale insects (flat brown lumps with slime undrneath) they hide mainly on the vain of the leaves and at leaf nodes but also on brown stems looking like bark
You cant do it with a spray no matter what the spray says, squishing by hand is necessary, also use a spray as there will bee millions of ones without shells that you can not see...a spray will easily kill these ones it kils thee ons with shells slowly as well , best spray is a teaspoon of neem oil from ebay dissolved in 1 litre of water with a tiny drop of dishwash liquid...it works better than any commercial spray i have tried.

when there is no strong sun spray the plant and also spray the neem on other plants nearby and thee sides and surface of the ground around
You also need to get rid of any ants fro on the plants , ants bring them and move them around, if you see ants on a plant they will bring scale and other insects