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I added a Model Input "Point Input" to my model. If I now launch the model from the Processing toolbox the user can click the ... icon and click on a point on the map.

How can I use the result of that in my model then? I tried making a buffer around it but the Buffer algorithm does not let me select the Point Input as Input layer.

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You can use the "Create Layer from Point" algorithm to create a usable model "object" from that clicked point:

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It's documentation (accessible if you open it in the Processing Toolbox (Ctrl+Alt+T)) states:

This algorithm creates a new vector layer that contains a single feature with geometry matching a point parameter.

It can be used in models to convert a point into a layer which can be used for other algorithms which require a layer based input.

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