Compromising integrity

Few days ago, I found I made a mistake in my work. I explained to the relevant party and admitted the mistake. However, it was not acceptable because the admission will means the customer may penalize us. For me, I feel we learn from mistake. Admit it and move on. However, to some party, admitting mistakes means weakness, and may jeopardize the dream to go higher and shine. The worst part was after I wanting to admit the mistake, I was told to write an additional line to my customer. This time it sounded like I did a check at that time, which I did not. However, it was insisted. I felt very bad because the person is practically asking me to paint another picture, and I feel it's an indirect lie. Anyway, I am trying to be as careful as I could because I need evidence that I am being truthful. Also, I wish the person who make me do this, good luck in the career, and hope the climbing up the ladder of success is smooth one. When the person stayed at top there just hope they will ask themselves, "Aren't you tired living like this?"