The District of Columbia school system is under investigation by federal and local officials based on allegations of test tampering.
The Highland Park district near Detroit is operating with a $11 million deficit, and may be placed under state control.
The foundation is expanding the number of judges who select the winner of the prize for high-performing urban districts.
The school board says that some top officials will no longer be required to live within the boundaries of the 157,000-student Texas district.
The results of the first year of a school turnaround effort in some of the district's K-8 schools shows an improvement in reading and math scores.
A teacher accused of abusing his elementary school students has prompted a deeper investigation into how the district handles allegations of misconduct.
The board replaced the policy, blamed for contributing to a school culture that lead to bullying, and eventually the suicides, of several gay students, with one that promotes "a respectful learning environment in which teachers facilitate student discussions of contentious topics in a balanced and impartial manner."
The measures follow the deaths of two children who had allergic reactions at school but were not injected with the pens, which can treat the signs and symptoms of an allergic reaction.
The superintendent of the country's second-largest school district said he wanted to bring in new staff members while the district and law enforcement officials conduct their investigations.
Current staff members at the school will be removed while the district and police investigate the charges against two teachers.