We have a circle of dead grass due to paddling pool being sat on it for weeks, there is a horrible smell from a part of it which has turned yellow! Is there anything I can put on it to help the grass or the smell?
A yellowed area of grass where something has been laying on it for longer than a week is to be expected; I would not normally expect the area to have a noticeable noxious odour, but clearly, if the paddling pool has been there for weeks (since May maybe?) then it's not really surprising.
So long as there's nothing obvious rotting on the surface that isn't grass, then wait to see if any of that area recovers and the grass grows again. Cut the yellowed grass; if it's too soggy, pull or scrape it off or do it with shears. If the area is compacted (which it very likely will be) then aerating it will help - if all you have is a garden fork and not a hollow tine fork, then insert the tines vertically to a depth of 4-6 inches and pull it straight out again, repeating this about 5 inches apart across the area. Grass will normally show signs of greening up quite quickly if it's not completely dead, but if nothing grows or shows signs of green by the end of August, and the grass in that area is clearly dead, then you will need to do some lawn repair. Given where you live, it's really too late to put down any fertilizer other than an autumn feed, but I would not recommend that at this stage - there's no point in trying to feed dead grass and its presence will not be helpful if lawn repair work is essential.
If the whole area where the pool was sitting is dead rather than just small areas of it, it's probably best to cut out the dead turf, fork it over and bring up to the level of the rest of the lawn with topsoil, level off and then sow seed. Otherwise, you might be able to just overseed, but in order to do the latter, you will need to create a layer of loose, friable soil for the seed to root into, and that can be difficult or impossible with a large dead area of turf beneath. Because of the cooler temperatures where you live, you will also need to sow seed very early in September, certainly within the first week, before it becomes too cold, and not use that area normally till next spring in terms of walking or any other activity on it. The other alternative is too remove the dead area of turf and buy a small amount of new turf to fill it in with; the drawback with this method is it is quite difficult to get the new turf level with the surrounding area of grass, but the advantage is, turf can be laid later in September and into October, will be growing by the time winter sets in, and will not be damaged if it is walked over occasionally (but only occasionally in the first six weeks).
For future reference, never leave anything on the lawn for longer then 7 (but preferably not more than 5) days, any more than you would on the plants in your beds and borders. It is easy to take the lawn for granted as just being some sort of carpet, but the grass plants in the lawn are just that, plants all growing together, and they need air and light to remain healthy.