the brown bits might be caused by a blunt mower. See if yours will easily cut paper. Or it could be leatherjackets eating the roots. Do you get birds pecking at it, and lots of long-legged flying insects?
give it a cut to level it off, and remove any remaining leaves or rubbish. You will have to box off the cuttings at this time of year. Dead leaves or wet mowings will kill or damage it.
if there are any bald patches, reseed them and rake it in.
You might benefit from a top-dressing to improve the texture. If you are on clay, lawn sand (grit) and sieved peat or fibrous compost will help. This will improve it more than a chemical feed. You have to brush it about well and apply after mowing so it does not lie on top of the blades.
Kids running about on it will work it into the surface.
If you are on sandy soil you need to add something heavier, but I have never experienced that.
Do you have a dog or other animals weeing on it?
If you really want to green it up, add a granular fertilser, but at the lowest application they say on the pack. This will just give it a temporary soft greenness and make it grow so you have to mow it. The effects will not last long and will make it soft and wet.
About 5:1 of lawnsandeat will be OK if you are on a heavy clay soil. The peat will rot or blow away within a year or so. You can add a little finely-sieved soil if it is not heavy clay but in your case it is not needed. Apply thinly and brush in. You can also use this to fill hollows and uneven patches, but never apply it thickly enough to bury the grass. Once the grass have grown up through it you can apply some more. It is better if the grass is short, tough and springy. If it is long or lush it will tend to be squashed down and buried and may die.
Apply any reseeding before topdressing and it will cover the seeds and prevent birds eating them.
BTW if you befriend next doors cats, they will look on your garden as their home and will keep it clean and not foul the lawn. They will also chase away other ill-mannered cats.
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