To clean, you should be using a soft, flannel patch that is twice the size of the pre-cut patch you might be using to load the gun with the ball. You can use plain water, or, in sub-freezing temperatures, Alcohol, as a lubricant, and solvent, to soak the patch before running it down the barrel. When the humidity is high, I like to run my damp( spit) patch down front and back, followed by a dry patch to dry the barrel completely, and pull out any moisture that remains in the corner of the grooves, or up against the breechplug. IN the winter time, I can sometimes get away with just using a damp patch to clean the barrel, and forgo the use of the dry patch. The alcohol will take moisture out of the barrel as it evaporates off the warm surface of the bore.
I found that using pre-cut patches to clean the gun with my JAG, was very unsatisfactory, because the patches have a finish to them, that seems to prevent them from quickly absorbing moisture and oils. They work to push the crud clean of the lands, but don't do as good a job as cotton flannel does in pulling the crud out of the grooves.
Oh, It is very important to have a cleaning jag of the correct diameter. I like the stepped jags, where the 3 or 4 " rings " are progressively smaller or larger, depending on which way you look at the jag.Use a caliper or micrometer to measure the actual diameter of the jag head, and of each separate ring.
You can step the jag yourself, by putting the jag in a drill press, and using a file to reduce the diameter of the soft brass rings. It only takes a few thousandths less diameter to help the patch blouse out and reach down into the grooves to remove the residues. Take the jag out of the rod, and use it to manually push a cleaning patch down into the muzzle of the gun. Then pull it back out slowly to see how the material blouses around the head. I believe that once you understand how the jag works, you will be able to use it properly in cleaning the barrel between shots.
The cotton flannel cleaning patchs work by " Blousing " out as the jag is pulled back out of the barrel. The folds get caught between the grooves, and the grooves in the jag, so that the cloth digs down into the grooves and pulls the crud out as you bring the rod back out. In general, pushing a cleaning patch DOWN the barrel clears residue from the lands of the bore: pulling the cleaning patch out(UP) with the jag clears the residue from the grooves.
If either my suggestion, or the suggestion to use TOW doesn't do it, get a bore brush and use it to push your cleaning patches down and back out of the barrel. Remember to turn the rod to the right( clockwise) to keep the screwhead tight as its turning in the grooves of the rifling, so the brush does not come off the rod down in the barrel. If the brush gets " stuck", just turn it clockwise, until the bristles bend back enough to go over the lands, and you will be able to " Un-stick it", and pull it back out. But always be turning it to the Right.