What does the phrase "其空有名头" mean?
其 is a possesive pronoun. The subject in the phrase is not mentioned.

空有= have only; 名头= title/name

空有名头 means "in name only" (either 'in a position that have no real power' or 'enjoy a reputation that is undeserved')

“其空有名头” = "his/her 'in name only'" 

We need more context to decide which meaning we are talking about.

Are you sure it wasn't '空有其名?' because  '空有其名'(only has it's name/ in name only) is a common phrase along with '虛有其表' (only has it's appearance/ in appearance only)