Can adenosine be cleared?
In order to do this you would need to find a compound that does all of the following:


Crosses the blood-brain barrier. Many pharmaceuticals are not able to do this.
Binds strongly to, or has some metabolic effect on adenosine. The resulting product must also cross the blood-brain barrier, and be excreted. 
Not bind or react with other molecules that are similar to adenosine and cause side effects.
Not cause some sort of immune response - along with the blood-brain barrier issue, this rules out any sort of protein, and really limits the potential specificity for items (2) and (3): there simply isn't a lot of chemistry outside of biochemistry that is that specific.
Not cause systemic effects via depletion of adenosine and products of adenosine. There are several problems caused by a failure of the enzyme that metabolizes adenosine but it seems like most of the primary symptoms are due to an accumulation of adenosine rather than an absence of inosine, so I am not certain how much of a problem this would be.


And probably numerous other issues I haven't thought of yet. The point is, adenosine is a useful biological molecule, not just junk to get rid of, and developing something to "clear" just one specific compound is actually really complicated. If the main concern is the sleepiness caused by adenosine build-up, caffeine is widely available and relatively safe compared to some novel way of actually clearing adenosine.

In fact, I can't really think of many pharmaceuticals that work by actually "clearing" something, but feel free to leave any suggestions in the comments. Chelating agents come to mind, but those are only used in some very specific circumstances.