NO. The TC Hawken breech plug is not designed to be removed. In fact, I have heard of people ruining their barrel trying to remove it.
IMO, the sure fire way to keep the breech clean is to use the advantage of the hooked breech your gun has to remove the barrel for cleaning it.
After removing the barrel, run a cleaning patch soaked with your favorite brand of cleaner (Dawn, Simple Green, Hoppes Black powder, Murphys Oil Soap....) up and down the bore a few times. Add a little water to the patch and repeat the up and down the bore routine.
Remove the nipple (with a nipple wrench) and lower the breech end of the barrel into a bucket of water.
Wet a cleaning patch with water and run it down the bore with a suitable jag on a cleaning rod. Pull it back up and it will suck water into the bore thru the breech plug.
I like to remove the breech from the bucket at this stage and then firmly force the patch/jag down the bore with a hard stroke. (This keeps the water in the bucket cleaner. It has also been known to dampen the spirits of neighbors cats who thought they were out of range).
A few up strokes and hard down strokes will blast away any fouling in the patent breech and the connecting hole between it and the nipple and the bore.
Change the patch and pressure flush it again, changing patches every few strokes until the patch comes out clean.
Some like to use a bore brush during this stage, but that is up to you.
Run some dry patches down the bore to dry it.
At this stage, I like to place the breech over the burner on my wifes gas stove to boil all of the hidden water out of the threads and passageways.
Lube the bore with a good gun oil like Sheath or Rem Oil.
Clean the nipple, oil the threads and reinstall it and your thru.