One word that was added to the prayer asking for forgiveness at the beginning of Mass, known as the "Confiteor" is the word "greatly." So we say, "I confess to Almighty God...that I have GREATLY sinned." In today's culture, we have "greatly" downplayed the existence of sin, and so have need to stand honestly before God and one another & confess that we are in fact, sinners, even great sinners.
The story about King David in the first reading is very much abbreviated. The slightly longer version is this: David lusts after a married woman and commits adultery with her. In order to disguise his sin to her husband who is in the military, King David tells this woman's husband, Uriah, to take a few days off to be with his wife. David is hoping that Uriah would believe the child is his. But Uriah refuses due to his military allegiance. So David has Uriah killed. Now he has sinned greatly by committing adultery and murder. The prophet Nathan confronts him on this and David repents, so God forgives him.
Jesus also forgives a woman known publicly as a sinner. When a Pharisee objects to this (at least in his own thoughts), Jesus tells him a parable about two people who were forgiven debts. One was forgiven a debt much larger than the other one. Which one is more grateful? The one who had the larger debt. So those who sin greatly are more grateful for God's mercy. Can we say with St. Paul, "I live my life with faith in the Son of God who loved me and gave himself up for me"? Do we, who sin greatly, have great gratitude and love for the God who forgives so generously?