The thing you can see is not the living part of the fungus, rather it is the fruiting body. So putting fungicide on it does not help, rather like putting weedkiller on a plant you have to treat the leaves not the seeds. In general putting fungicide on lawns for fungi growing in the soil is not reckoned to be effective. But it is possible that this is actually parasitising the grass itself, I'm not sure.

The particular object you have growing is familiar to me, but I'd have to check some books at home to remember the details. I've not seen it in a garden before, but I normally see it on poor, acidic soils. I think if your lawn was in general healthier, as juicer says, you'd get less of it.