How do I apply an effect only to a part of an object?

Not sure if this is possible: I want to apply an effect only to a part of an object and specifically the part that I'm hovering over with the mouse cursor or the touch input.

It depends on which effect you want to use.

You need another sprite as a mask (white circle for example), put both sprites on the same layer (mask should be on top).

Try applying different effects and/or blend modes to the mask sprite.