A mouthhabit is something you say with your mouth a lot. It's a habit. It's your favorite phrase or catchphrase. Many people have the habit of saying the certain things over and over again, whether they want to or not.
The reading is the kun'yomi vocab reading of both kanji. 癖 is the reading you learned with the kanji. Watch out for the rendaku.
That chef’s catchphrase is, "Such and such is in season."