I'd purchase or hire a snake, feed it into the tube and pull out to extract any gunk that is in there. If that fails you could attempt pressurizing the tube, but be aware of any sealing that could fail.
The first thing you would have to do is take out all the kitchen stuff from under the clog tube so that when we are working on it dose not make a mess. Then you will need to take out the tube from under the sink so you use a water pressure to unclog all the stuff inside the tube.
