The priest who was already ministering to Italian immigrants, Fr. Morelli, was not the mindless idiot shown in the movie.
Archbishop Corrigan was not the son of poor working class Irish immigrants in New York City. Corrigan was born in New Jersey to Irish immigrant parents who owned a retail grocery and liquor business in Newark and were well-to-do.
The nuns did not spend their first night in a brothel..Archbishop Corrigan had temporary lodging at another convent ready for them when they arrived.
When Cabrini is talking to the mayor, the letters on the flag are backwards, indicating that the film has been flipped.