Add custom field to Posts and sort by it

The easiest approach would be to use the Advanced Custom Fields plugin.

A quick example for your situation (this example assumes you have access to your theme files, and can update the PHP for output of the 'Scores' field):

1. Install and activate the plugin
2. Navigate to Custom Fields in the left-hand nav
3. Create a new Field Group
4. Create a 'Scores' text field
5. Output the 'Scores' field in your page/post
6. Use WP_Query to load and sort the posts by your new 'Score' field

Good luck!