What is the expiation for throwing a live cat into a flowing river with the intention of killing it, and is it permissible for her to give charity or fast on its behalf if she does not own any money?

A cat may not be killed unless there is a Shari justification. If there is a justification, it is still not permissible to kill it if it is possible to get rid of it by another means. The method used to kill the cat contradicts the saying of the Prophet (peace be upon him): "Indeed, Allah has prescribed excellence in everything. So, when you kill, kill well." The person who did this must repent to Allah Almighty. There is no specific expiation for this act other than repentance.