Well, before inserting a new score in the table you should make a query to check if in the table already exists a record with the given name, in that case you'll have to make an update on the existing row instead of an insert.
But there are some things that you may want to consider:
1: are names like "John" and "john" the same or should you consider them different?
2: is really the name a proper index to save a player score? What if two different players use the same name? Their scores will be merged into one! Would it be correct and acceptable? If not, we you should study a way to tell these players apart