You Do Need power plug adapter while traveling from Channel Islands (Guernsey & Jersey) to Cook Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cook Islands is I and in Channel Islands (Guernsey & Jersey) are C and G. This is how they look like.

Channel Islands (Guernsey & Jersey)

In Channel Islands (Guernsey & Jersey), outlets of type C and G are in use.

Type C

Type G

Cook Islands

In Cook Islands, outlets of type I are in use.

Type I