All the exceptions to the hearsay rule would apply regardless of the nature of the case. There are few cases in which special circumstances dictate a special exception to the rule, primarily instances such as "dying declarations."
Child abuse cases are troublesome as the child often does not understand what is going on and may parrot remarks by an adult in order to please the adult and/or avoid perceived punishment. Although one would prefer that a child not testify to abuse, it would be difficult to obtain a conviction without the child's testimony. Although not a true "exception" to the rule, the Hearsay rule does not apply unless the uttered statement is made in order to prove the truth of the matter asserted. If offered merely to prove that the person made the statement, it is not hearsay and not covered by the rule.
Two notable exceptions that should apply here are the Res Gestae or "excited utterance" rule. If one makes a statement during the heat of the moment, immediately after a serious or traumatic event, an exception applies. The reasoning is that one has not had a proper opportunity to amend, correct, or "perfect" his story and it is therefore probably true. A second exception is the "admission against interest" rule; that is a "confession." Thus if one admitted to an improper or unlawful act to a second person, the second person could testify as to the truth of the statement as an exception to the rule.
Hearsay is rarely admissible in legal testimony because it is generally considered unreliable and often inaccurate. But there are some important exceptions involving what people tell medical personnel when seeking treatment, and in child sexual abuse cases.
In the case of the abuse of children, the exception exists for several reasons. Psychologists and sociologists have argued that children are less likely to lie, especially about abuse, because their speech and thought patterns are not yet fully developed, and they are still learning how to communicate. This does not lend itself well to misinformation. Also, child abuse cases sometimes take years to come to trial, during which time the child may have forgotten the details of events that happened when they were younger.
These reasons are why hearsay conversations with medical personnel, therapists for children and statements about the abuse made to relatives are most often considered hearsay exceptions in legal testimony.