Jesus does not mince words today regarding the importance of avoiding sin. He says plainly in this familiar Gospel passage that we should avoid sin at all costs. Considering that sin is a turning away from God, his insistence seems appropriate. Sin fundamentally goes against a life oriented toward God.
St. Ignatius, in his First Principle and Foundation, also asserts that we ought to direct our lives to God in every way. This might include “cutting off” something that causes us to sin (a bad habit, a harmful attitude, etc.). However, I think the same urgency could be associated with “putting on” something that brings us closer to God (spending more time in prayer, giving of ourselves to others, etc.). I could imagine Jesus saying in the same breath that if there is something in your life that causes you to be closer to God, then by all means do it!