Phoenix will definitely win!!!

She is a Goddess!!!, The power of the sun herself!!!
She as the ability to manipulate superman's thoughts at will.

Superman cannot defeat her because she is a million times better than him in Strength, Intelect, Speed, Agility, and EVERYTHING!!!
Even though the sun provides energy to superman, the power she possesses is controlled by her and definitely will not supply Superman with energy!!!
Pheonix also has some of the abilities that Professor Xavier has and can probably kill people by just thinking of them. Phoenix DOMINATES superman because he can die in less than a minute as Phoenix thinks about his death.