If I agree to sole custody awarded to my son’s father, can my son's father tell me that my boyfriend is not allowed around my son?

The answer to that question depends upon the reason your son's father would want to prohibit your boyfriend from being around your son. If your boyfriend has any history of child abuse or domestic violence, the court would probably agree with the prohibition. You need to consult with an attorney before you agree to sole custody to the child's father; the best approach would be to include some specific provisions in the custody agreement that preserves your ability to have your boyfriend around when the child is with you. If there is no agreement, the matter can be referred to mediation or even a court hearing to get a decision.