fight over (someone or something)
To argue about who gets access to someone or something. Are the kids still fighting over that one toy? Yep, I saw two people fighting over a TV on Black Friday.
fight over someone or something
to fight a battle that decides who gets someone or something. Well, let's not fight over Tom. You can have him. I can make a better team without him. The children were fighting over who would get the largest piece of cake.