If the solder readily sticks to the trace, you have removed the corrosion. If the solder does not want to stick to spots and the area turns dark colored, it needs an abrasive cleaning.
I abrasively clean the traces down to clean solder or clean copper. If the battery damage is severe, it is possible it has corroded all the way through. I always solder tin abrasively cleaned traces.
If a trace is blown out and the gap is small, i use a resistor lead or something appropriately sizes to bridge the gap and tack it down to the trace. Other times a fine wire appropriate. I try to avoid using long wire runs if all possible.
Your work is looking pretty good so far.