Easy way, Hire a spiking machine that takes out lots of little plugs. Fill the holes in afterwards with sharp sand/grit.
The best way to do it is to take up the lawn. Rotovate a lot of compost (3 inch depth worth) into the top nine inches,then grit as well into the top three inches, and then reseed or relay lawn.
You need to think about a number of things before you invest a lot of effort and money. First, just how waterlogged and for how long. If it is only a minor, brief problem, take fidgetbones' advice on the easy way. Then you need to consider where the water is coming from and where you want it to go to. If you are at the bottom of a slope and surface water runs down into your garden, raising the level will not help. If there are underground springs or a leaky water source, soil improvement will not help. If water pools in your garden, you need to find somewhere for it to go to and a means of directing it. So, a complex problem. Sometimes, heavy clay just need a bit of improvement, but the hundreds of threads on here about drainage will show you that you need investigation before action.
The lawn is quite large, although the area affected isn't, but it happens in more than one spot. Would laying compost or well rotted manure onto of the grass help, or would it just become a mud bath?
It is only when it rains. Problem is Fairygirl, I think the water table is so high, that it doesn't take much rainfall for it to appear on the surface.
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I live in west central Scotland - not where that photo is...
I live in west central Scotland - not where that photo is...