It depends on what part of the play that you are referring to. In Act Two, he has information about the accusations that are occurring that could possibly expose the entire proceedings in the courts as erroneous. Abby, in act one, told John that Betty was not sick because she was bewitched; rather, "she took fright is all," because her dad caught them dancing in the woods. Abigail is one of the main accusers now, and Betty herself claims to have been bewitched, and has accused others of bewitching her. The information that John has exposes the girls as liars and frauds. Elizabeth want John to go to town and tell them this information, to try to set the accused women free.
John hesitates to do this for a couple reasons. The first is that when Abby told him this information, he was alone with her; so, he has no witnesses to what she said. And, he wonders,
"If the girl's a saint now I think it not easy to prove she's a fraud."
The town, at this point in the play, loves Abby and believes everything she says--she is hailed as a saint and savior to the town. If John walks in there and tries to prove she's a liar, based on something Abby said that no one else heard, it's going to be hard to prove. So, he hesitates for that reason. Another reason he might hesitate is one that Elizabeth refers to, that he was once very close to Abigail, and he doesn't want to hurt her or make her look bad. There may be lingering emotions.
Later in the play, John hesitates over the issue of whether to confess to witchcraft or not because he knows it would be a lie, but, he doesn't feel like he is a good enough man to die with integrity. He wants his life back, but doesn't want to live a lie either; however, he doesn't feel comfortable dying for virtue when he has been a sinner in his life. So that is why he hesitates at the end.
I hope that those thoughts help a bit; good luck!