How to use mysql LIKE with wpdb?

According to the Codex, the delete method will not accept the LIKE operator. You’ll be better off just using the query method.

$query = $wpdb->prepare("DELETE FROM $wpdb->posts WHERE post_type="flamingo_inbound" AND post_content LIKE %s", '%'.$mail.'%');

$wpdb->query($query);

Hopefully $mail doesn’t have a % or _ character otherwise you need to escape it before you build $querylike so:

$mail = $wpdb->esc_like($mail);