Uh-oh Aaron. His girlfriend has been granted immunity and will be able to testify against him for the prosecution.
It's a little different than what your bare statement suggests. The reporting is spotty and we'd have to see the motion and the ruling to fully know what's going on here.
However - without immunity Jenkins, if called by the prosecution, could "invoke the fifth amendment" and refuse to testify to anything that might result in self-incrimination. That is, she wouldn't have to testify to any bad behavior on her part, as long as it was possible
for the state to charge her with something based on that, or, as long as it was possible
for the state to use her testimony against her in a future matter. Say, for destruction of evidence. If her refusal to testify also somehow shields AH, that's just a collateral issue from a Constitutional perspective.
But now, if Jenkins has been granted full or limited immunity, she can no longer invoke the fifth regarding any question the immunity addresses. That does not mean she will tell the truth in response to any question, but simply that her response can't be used in a future proceeding to incriminate her (say, distraction of evidence).
If she testifies and lies, she can still be charged with perjury (should the state wish to charge her so.) If she refuses to testify, the Court can hold her in contempt, and, depending on the motion/order/MA law, she might still be charged with a crime (say, destruction of evidence) if she refuses to testify on that subject and/or perjures herself.
In my experience these kind of agreements can be tricky and depend on some measure of goodwill. Should one side start to game the other, there's usually some penalty for it. Usually though, the safe path is for the immunized person to have every incentive to spill their guts (overprotection) and some strong penalties for game-playing. Jenkins might have anything ranging from partial immunity (nothing she says is useable against her, but an independent case might be made) to total immunity (if she testifies, no charges can be pursued against her.) We'll have to see.