To clean the cover plates properly, remove them from the harmonica. Even just rinsing them in warm, soapy water and a light brush will clean them up pretty well. Dry and wipe down with a soft cotton cloth or rag. Don’t soak the whole instrument in water. You can wash plastic combs. But do not wash wooden combs.
Then after using the instrument, a good way to clean and shine the instrument, is to use a silver cloth as used by orchestral/marching bands on tubas, cornet, horns etc. These cloths are made for silver plated instruments and are an excellent cleaning tool for harmonicas. Trevor James (flutes) and Denis Wick (mutes and mouthpieces) are two manufacturers who make these cloths. Both are excellent quality.
Finally, just wipe down the outside of your harmonica after playing and that will help keep your instrument clean and in good condition. It’s much nicer to play an instrument like that.
I hope that helps.