There are two types of cursor, hardware cursor and software cursor. You are probably using the software cursor to change the cursor texture. Software Cursors visually lag a few frames behind the actual position of the hardware cursor, which you use to click things. You might want to configure a software cursor for every cursor state possible in the project settings so that this is not noticable and so you can have every cursor type in your game style. Alternatively you should also be able to customize hardware cursor textures but I have not tried that.
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