Cleaning your mohair tack is actually quite simple!
Mohair cinches (and breast collars) need only be washed when very dirty.
Start by using your hands or a soft brush to remove any debris and heavily caked on dirt. You do not need to remove horsehair as it will act as additional padding.
Next, dunk and swish your cinch in a bucket of cool or room temperature water. Never use hot water as it can shrink the mohair and cause colors to bleed. You can add wool lite or a similar wool soap to the water.
Rinse your cinch with clean water from a new bucket or the hose.
Leave it flat or hang to dry. Keep it out of direct sunlight as this can also shrink the mohair and cause the colors to fade.
Never put your cinch in the washing machine - this can damage both the cinch and the machine. Typically, you should only need to clean your cinch once or twice a year.