I clean cables and housings by removing the cable. Wipe down the cable to remove any gunk on it. If it shows signs of corrosion or has any broken wires, replace it. I cleam out the housing by using a spray can of lightweight lubricant. Press the straw tight against the housing and spray. the force should blast any accumulated dirt out of the housing and leave a film of lubricant behind. Reinstall the cable and adjust the derailleur.
Usually it's the rear derailleur cable that has problems, specifically the loop next to the derailleur. But, it's worth checking the front cable before doing anything more drastic.
Once you disconnect the cable from the front der, remove the chain and operate the der by hand. It should be fairly hard to move because of the spring tension and should easily return to the inner position.