First and foremost, your priority is to your daughter's safety, even above hurting her feelings. As parent's, we often have to make many decisions that our children don't like at the moment, but WE know is better for them. That said . . .
1. If he truly can't be trusted, you will need to find a way to tell your daughter visits will only be when he can come to your home (or not at all, or whatever you decide). And you will probably have to take some sort of legal action as well.
2. If you can give him another chance,
Can you have a heart-to-heart with her father? Explain to him clearly that you are concerned about the aunt's boyfriend, and under no circumstances should your daughter be allowed there without his direct supervision. Explain your daughter's desire to spend time with HIM, and suggest that if he wants to go out on his weekends without her, you are happy to come pick her up and cut their visits short.
Work together to lay-out guidelines as to what is acceptable during the visits (a good time to get on the same page on discipline too), and state consequences from the beginning. For example, if she gets dumped onto the aunt and grandma, no more visits, period. Could you also provide your daughter with her own phone just for the visits (no, I am not otherwise advocating kids have cell phones)? This way she could call you the moment she is dropped off at the aunts home alone and you could come pick her up.
Bottomline, your daughter's safety is most important! Work to protect her! I pray you will have wisdom in handling this situation . . .