We study a general class of repeated auctions, such as the ones found in electricity markets, as multi-agent games between the bidders. In such a repeated setting, bidders can adapt their strategies online using no-regret algorithms based on the data observed in the previous auction rounds. Well-studied no-regret algorithms depend on the feedback information available at every round, and can be mainly distinguished as bandit (or payoff-based), and full-information. However, the information structure found in auctions lies in between these two models, since participants can often obtain partial observations of their utilities under different strategies. To this end, we modify existing bandit algorithms to exploit such additional information. Specifically, we utilize the feedback information that bidders can obtain when their bids are not accepted, and build a more accurate estimator of the utility vector. This results in improved regret guarantees compared to standard bandit algorithms. Moreover, we propose a heuristic method for auction settings where the proposed algorithm is not directly applicable. Finally, we demonstrate our findings on case studies based on realistic electricity market models.