It was traditional in the past for established geisha to take a danna, or patron. A danna was typically a wealthy man, sometimes married, who had the means to support the very large expenses related to a geisha's traditional training and other costs. This sometimes occurs today as well, but very rarely.
A geisha and her danna may or may not be in love, but intimacy is never viewed as a reward for the danna's financial support. The traditional conventions and values within such a relationship are very intricate and not well understood, even by many Japanese.
While it is true that a geisha is free to pursue personal relationships with men she meets through her work, such relationships are carefully chosen and unlikely to be casual. A hanamachi tends to be a very tight-knit community and a geisha's good reputation is not taken lightly.