I am an ATR (Absent Teacher Reserve) and have been assigned to a school a far distance from my home. Can I file a hardship transfer request?

As an ATR, you should apply for an "accommodation" from the DOE. More information and the form needed can be found at: https://www.uft.org/files/attachments/personnel-memo-4-2009-10-disabilities-act-accommodation.pdf. 

However, regularly appointed teachers who are in a school very far from their home would apply for a hardship transfer.