In Marvel's past crimes as the Winter Soldier, as seen in the new series on Disney Plus.
In the premiere episode, it's revealed that Barnes has received a pardon for his actions as HYDRA's brainwashed assassin known as the Winter Solider after the events of Avengers: Endgame. However, the pardon is conditional upon the agreement that he attends therapy sessions in order to closely monitor his mental state. In a session with his therapist Dr. Raynor (Amy Aquino), it's revealed that Bucky has a notebook and list of amends his been attempting to cross off, all while following the three steps he was given to follow by Raynor to help with his ongoing recovery: Don't do anything illegal, don't hurt anyone, and tell the subject involved in the atonement that he is no longer the Winter Solider, he is James Buchanan Barnes, and that they're part of his efforts to make amends (all with a smile).
Regardless of his inability to maintain Raynor's first two rules as strictly as he claims he is, Bucky is still trying to cross names off his list with deliberate and proactive intention for specific harms he's caused. The same can't exactly be said for other Avengers such as Black Widow who had "red in her ledger" or Wanda Maximoff and the accidental deaths she caused in Lagos at the beginning of mistakes she made in WandaVision with the pain she caused to the town of Westview. Rather than taking the generic method of trying to be a better person and balancing the scales in general like these other Avengers, Bucky is actually being purposeful and intentional in his journey for redemption, despite the hardship and emotional turmoil that inevitably comes with it, as evidenced by the premiere when he's struggling to confess to murdering a man's son.
When Steve Rogers' Captain America was thawed out and awoke in the present, he wrote down a list in a notebook of all the fun things he needed to catch up on such as movies, music, foods like Star Wars, Thai Food, and Marvin Gaye's Trouble Man soundtrack (per Sam Wilson's recommendation). Sadly, Bucky Barnes doesn't get to have that luxury, as his notebook and list are much darker than that of his longtime friend's. However, seeing Bucky make these attempts to redeem himself will hopefully allow him to sleep better without any PTSD-driven nightmares of his Winter Solider past, allowing him to move forward with his life simply as Bucky. Furthermore, it's incredibly commendable and a testament to his character that he's trying to truly atone for actions, as it's a lot more than any of the other Avengers can say in regards to their attempts to redeem their sins and mistakes.
In any case, Bucky Barnes will no doubt have a big chance to atone for his crimes even further, especially with the knowledge from trailers and teasers that the villain Helmut Zemo will soon be involved in future episodes of The Falcon and the Winter Soldier. Perhaps when the series ends, Bucky will finally be able to fully atone for his past as the Winter Soldier once he's able to cross off all the names off his list of amends.