score:8
The question is quite complex to google. Per the comment I dropped, the edit that added the detail seemed somewhat suspicious at first glance due to the lack of citations.
After digging a bit deeper though, I found a few other corroborating sources, including a few in Google scholar, and a book by Bruno Mugnai and Alberto Secco (La guerra di Candia 1645-69, in Italian), so my assumption is that the information is legit rather than pure invention.
The reward seems to have been the fort of Temenos if this article (or this one, which has the exact same quote) is anything to go by:
During the Cretan War, i.e. the war between Venetian-Cretans and Turks, most battles took place in the countryside, as the Great Castle of Candia resisted for 22 years. In the wider area of Heraklion, Christians were led by the abbot of the monastery Agarathos, Athanasios Christoforos and the scholar Gerasimos Vlachos, while the Venetian general Gildasis (Gil d 'Has) attacked the Turks in several places. During one of these attacks, in 1647, Gil dā Has attacked the Turks who had occupied the fortress Temenos, slaughtering almost all of them. In 1669, after the Turks conquered Candia, the fort was donated by the Sultan to the Venetian traitor of Candia, Andrea Barozzi. The fort was then named Kanli Castelli, i.e. bloody fort, so as to commemorate the massacre of the Turkish army in 1647. However, there were references to the fort, naming it as Nefs Temenos.
I was unable to locate why Barozzi defected however. That said, the answer might be in this book. If the only comment is anything to go by, it is a first hand witness account of the events through the eyes of a French officer. (The siege ended shortly after the French failed to relieve the fort.) It turned up in a Google search with Barozzi as a keyword, so with a little luck a few of the documents might give hints as to why Barozzi switched sides.