The storm left,having caused a lot of damage to this area.

1.它应该是结果状语.结果状语一般分为两种:
用Ving的表自然而然,顺理成章的结果;
用不定式的表出乎意料的结果.
如:Many fans rushed to the airport,only to be told that the superstar had left eariler.
Soccer is played in most countried in Europe,making it the most popular sport.
2.此处用了Ving的完成形式,同样表示动作的先后,即先caused a lot of damage to this area,然后再left.