Yes, I believe she does because she says she does. I can’t imagine saying it otherwise.
I don’t think it’s about his soul, though, and certainly not about sympathy or absolution. I think it’s for his wisdom, guidance, and well-being here and now; for him to be led in the right path and serve well the nation that has been entrusted to his care. If I were still a praying person, I would pray for that too.
I grew up among people of sincere religious belief, and they did pray for the president, with genuine feeling, whether he was elected in 1952, 1960, or 1968.