Do Not Seek Absolution is a multi-part side mission available in Red Dead Redemption 2. Arthur wants to help Edith Downes and her son Archie in achieving a quiet life.
Additional note - This mission allows you to increase your honor (reputation) to the maximum level. The quest will only appear when Arthur has enough positive reputation. If you made "bad" decisions, try increasing your honor by, i.e. helping people in random encounters or by making honorable decisions in missions that are part of chapter 5 and 6.
The first chance to unlock this side mission appears in chapter 5 (you can also start it in chapter 6). To trigger it, Arthur must visit the doctor in Saint Denis, and have a high reputation (honor). After that, the game will add a quest marker in Annesburg. Find Edith Downes - the woman is earning money by working as a prostitute.
Watch the cut-scene. Go to the mine located west of Annesburg. A few miners are harassing Mrs. Downes's son. Intervene - beat the foreman. As always, remember to use blocks and counterattacks.
Wait for the second quest marker to appear in Annesburg. Find Archie (he is standing between the buildings). Start looking for his mother. Follow the path that leads north-east. You have to reach the path that goes under the bridge. Activate the Eagle Eye - notice the blue track. Start following it.
The tracks start on the main road, but soon they will direct you into the wilderness. Keep using the Eagle Eye to be sure that you are following them. Reach Mrs. Downes and a man that is accompanying her. Defeat that man. Speak with Mrs. Downes - we recommend choosing the Ask option (Square / X). Go back with the widow to Annesburg. Arthur will give Mrs. Downes and her son a bit of money.