【Updated: 】
Some skills with healing effects that staff-wielding Heroes have do not allow them to receive experience or SP.
Please check the skill's description to see if it says that the Hero cannot receive experience or SP. If it does say so, try equipping the Hero with another healing skill that does allow them to receive experience or SP.
Skills from which Heroes can receive experience and SP: Reconcile, etc.
Skills from which Heroes cannot receive experience and SP: Rescue, Return, etc.