A noise gate is a somewhat passive function that is made to cut a continuous noise floor from silent parts. Low quality mics emit such a noise, or the fan of your computer. If the volume of the source is below the noise gate threshold, the output ist turned off, so the noise isn't heard while you are silent. If you speak, the volume is above the threshold, and output is opened, so you hear the voice - and the noise, but since the voice is supposed to be much louder than the noise, that's usually not perceivable.
There are also noise suppression filters available in OBS, they process the audio to try to remove the noise floor with digital signal processing.
To split a click noise from your voice while you speak, advanced signal processing is required. The click has to be computed out of your voice. Such filter doesn't come with OBS. All these signal processing filters also distort your voice, so they are only last resort. Best practice is to not record mouse clicks in the first place. Your mic should be much nearer to your mouth than to the mouse.
The Blue Yeti is able to pick up in a directional pattern (cardioid mode), so make sure you activate the corresponding mode, and your mouth is within the pickup zone and the mouse isn't.