Cut out the area where the cement board has suffered damage. Attach the appropriate blade to the angle grinder. Push the blade down so that it will cut into the board and then pull the trigger. Hold the grinder with both hands and cut down into the board. The average depth for floor Hardie board is a 1/4 inch and for walls it is 1/2 an inch, so cut accordingly.
Use the pry bar to pry the pieces of broken Hardie board out of the way and scrape the floor free of thinset mortar until you are back down to the base layer beneath the underlayment.
Measure the area and cut a piece of new Hardie board to replace the old. Cut the piece with the angle grinder. Spread a layer of thinset mortar onto the installation area with the notched end of the trowel and push the piece of board down into place. Nail it every three to four inches to secure it.
Cover the seams with mesh tape, such as what is used for installing drywall. Take the remainder of your thinset mortar and use it to mortar the wire mesh tape to the face of the board along the edges of the joints with the flat side of the trowel. Let the repair dry for a minimum of 24 hours before installing tile on top of it.