How do I correct a name for a golf course on the map?

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I'm guessing that this is the node that you added. If so, it's not showing as a golf course because it's not labelled as one. There is a tag "leisure=golf_course" that does this.

As you note above there's isn't an icon available to add to nodes to label them as golf courses. However, if you were to draw a way around a golf course and then label it you'd see "golf course" available under sport and leisure.

You don't need to do this, though, because someone's already mapped the golf course here - they've just misspelt the name. So if you edit your node by clicking here, zoom in a bit to make it easier to see what you're doing, delete it, and then move up a bit towards West Chester Pike, you can see the green outline of the golf course. Click that, and you can correct the spelling.