I know this is Just Another Image Uploader Problem thread.
The uploaded file could not be moved to /var/www/vhosts/domain.com/httpdocs/wp/wp-content/uploads.
I have to chmod many directories to 777 to get uploads to work, so I chown'ed wp-content/uploads to apache:apache, which is what WP-created files get as owner:group (like wp-config.php).
So I chmod'ed it to 777.
So what we have is the server trying to write a file to a 777 directory it owns, and it can't.