The term CUTS INTO can be solved by two crossword solutions, ERODES (6 letters) and INCISES (7 letters). ERODES is a verb which means to gradually wear away or cause damage to something, usually by the action of wind, water, or other natural forces. INCISES is also a verb which means to make a cut or incision into something. Both words describe the action of cutting into something, but ERODES implies a gradual process while INCISES implies a more sudden action.