Yes,
Just use the brass jag that comes in the pack w/ the gun. You can either buy patches or go to the cloth department at wally world and find some fabric on sale for 1 buck a yard. Find some good 100% cotton fabric and buy a couple of yards. Cut it into squares and for 2 bucks you have all the patches you will need for awhile.

The junk you saw in the breach area would be removed by the jag/patch/spit combo.
Some folks use diluted windex or something similar. That works just as good as spit. But I find it easier to carry around spit than windex

Hank