I clean my Mossberg pump every time I use it. First, remove the barrel, this will simplify things no-end.
I use "Youngs" oil, which is diluted 1 part oil to 3 parts water to form a white emulsion. Put a patch soaked in this emulsion on the plastic jag, and shove it up the bore and out the muzzle. Next take a bronze brush and vigorously scrub the bore, especially the forcing cone and choke area. Then use a succession of clean patches soaked in the oily emulsion on the jag and push them through the barrel until they emerge from the muzzle relatively clean. Dry the bore by pushing several patches through, and finish by passing an oil-soaked patch through the bore. When I say soaked I don't mean dripping with oil - there should be just enough to leave a thin film of oil on the bore. A cleaning kit usually includes a wooly brush for this purpose.
Spray the outside of the barrel with the red can of Browning 'Legia' oil (or similar product), leave for a few minutes and wipe down with a clean cloth.
It is important to vigorously scrub the forcing cone with the bronze brush - plastic builds up in this area and will lead to rusting of the bore under the plastic.
Last week I bought a semi-auto and the barrel is permanently fixed to the receiver. Thankfully this gun has an IC choke, so it was quite easy to clean from the muzzle. I noticed plastic build up after only 250 rounds.