Yeah, I don't know about washing carbon... maybe it works, have not tried it... but with the way carbon filtration works I don't see that happening... If running water through it would clean it then everyone would be doing it. Also, if running water through a carbon filter worked then why do they make water filters with carbon filtration? They make them that way because carbon works like that... water/air passes through, it fills up, it's done.
I would think that if anything would work it would be some kind of soaking of the carbon. Using something that separates the carbon from the dust. What does this? I don't know.. Carbon works by having a HUGE surface area for such a small piece. The surface area is made up of crevices and cracks that are very very small. As air/water/etc passes through the carbon, small particles get stuck in the cracks and crevices and a more pure air/water passes through. With that in mind... I ask, how running water will do anything for it?
wishful thinking... I hope it does work for you though. If it does, let us know.
Still... i think the first response (mine) was the only thing that is going to work for you for more than maybe 1 month of poor filtration... buy a new one. (or replace carbon)
If you are going to replace it... make sure you pack it very very well. In manufacturing, the units are shook/vibration to make sure the carbon falls into the most compact arrangement... This would be hard to achieve without the right equipment.