Puncheon has spent the first half of the season at Selhurst Park on loan from Southampton and will now move to the club on a four-and-a-half-year deal.
Hennessey arrives from League One Wolves on a three-and-a-half-year contract and will challenge Julian Speroni for the number one spot.
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"I was here when I was eight years old when the likes of Dean Gordon and Chris Armstrong were here, so that makes it special.
"The massive factor in this was the chairman and the manager, the gaffer has shown great faith in me since he's been here and that's been unbelievable for me.
"It's a boost for me mentally - I thought it might happen at the end of the season but it's come quicker and obviously that's good for both me and the football club."
Hennessy, who has Premier League experience from his time with Wolves, added: "I'm delighted to be here, I've met some of the lads and the staff already so I'm just looking forward to getting out onto the training pitch now.
"We've got Arsenal on the weekend which is going to be a tough game but hopefully we can get something out of it."