Don't worry about soil being boring to chickens, it won't be. They love scratching in it and dust bathing in it. It's even better if it has grass or weeds growing in it. Don't level it for the benefit of the chickens, they can manage getting around quite well. If you level it you will be destroying the turf.
All that growth in the back is a bit of a dilemma. One one hand they'll enjoy playing and hiding in it and looking for fun creepy crawlies to eat but they could easily hide a nest in there too. I'd probably clean it out enough that I could move around in there and spot nests but leave them enough shrubbery that they can hide from aerial predators.
In the UK you have a pretty mild and somewhat damp climate, grass should grow pretty well. That area of grass does look like it is small enough they will denude it but I like to see what will actually happen instead of just assuming. I'd at least let them have the fun of denuding it and see how wet if gets before I did anything with it.
If it stays dry then bare dirt isn't bad. In the UK it probably won't stay dry. It's much worse if water drains to that area and stands. Walking around in the mud is inconvenient and messy, you don't want that. But the real problem is that if it stays wet for a few days it can start to stink and become unhealthy for the chickens. The microbes that break chicken poop and organic materials like wood chips or shavings, hay, straw, dead leaves, and such become anaerobic. Those stink and become slimy, not pleasant at all, let alone the health issues. For me when I'm locating a coop or run drainage is the main issue.
I'll include a link to an article about fixing muddy runs that might be good for you to read. Hopefully you won't have any issues, at least not bad.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/articles/how-to-fix-a-muddy-run-chicken-coop.47807/
Many people are able to handle a muddy run by putting in organic matter. If the bugs breaking it down go anaerobic and it starts to stink they may have to remove it and replace it but it does often work. If you have a problem it's worth a try. Also you don't have a lot of chickens for that area so the chicken manure should be spread out during the day. If it doesn't build up thick the manure isn't usually a big part of the problem. That's a good thing about having a run larger than the absolute minimum. They don't move around at night so the poop can build up under the roosts, that may be the only place you really need to manage the poop.
The deep litter method is where you turn your coop floor or run into a compost pile. I don't know what your coop looks like, if it is on the ground with a dirt floor you can do that. If it is an elevated coop you probably don't want to as that can rot the wood. For it to turn into compost the bugs need enough moisture to live and reproduce but you don't want it to get so wet that they can't get air and turn anaerobic. My coop stays so dry those microbes can't live so my method is not the deep litter method.
If you turn your coop into a compost pile you may be able to mix the poop under the roost in with it, either you mixing it manually or toss some scratch on it so the chickens will mix it for you with their scratching. I use a droppings board and put that poop in my separate compost pile. If you turn your coop or run into a compost pile you can put anything in that you would put in true compost pile. Kitchen scraps, garden excess or scraps, dried leaves, grass clipping, whatever. With chickens wood chips can work but I try to keep wood products out of my compost pile, they take too long to break down. I want that compost. I think whatever you use should be inexpensive and plentiful. Some people bag dried leave in the fall and use those through the year as needed. Grass trimmings can work well. Sometimes you can get a utility to drop off a load of wood chips where they are trimming trees near overhead lines. Some municipalities offer wood chips either free or not very expensive but you usually have to haul them yourself. It might be worth a call.
I really hope you don't have any of these problems, many people don't. Or the fix is as simple as spreading some type of bedding. We all have out own unique situations so we get different results. Sometimes trial and error is necessary. So good luck, that should be a fun area for them.
