What prayer should be performed to facilitate marriage to a girl whose mother opposes it, and is it Salatul-Istikhara or Salatul-Hajah, and what are the ways to facilitate this matter, especially since the girl felt distress and shed tears after performing her Istikhara?

Salat al-Hajah (definition) (the Prayer of Need) is prescribed for facilitating permissible matters, and it is used to seek ease in achieving a legitimate purpose. There is no specific act of worship mentioned in the Sharīʿah (Islamic law) for facilitating marriage, but supplication (duʿāʾ) is beneficial for every legitimate request. As for feeling distressed after Salat al-Istikharah (the Prayer of Seeking Guidance), this is not what should be relied upon. The preponderant view is that a person should proceed with the matter afterward and not abandon it unless Allah diverts them from it.