However, regarding the cage... There are several elements to consider:
1) You don't say how large or small this cage is; however we are assuming it is a small crate type enclosure. Metal wire cages are just terrible for dogs as the wire grating can cause havoc on paws. And both small crates and cages are too confining for a dog to be in overnight.
For your Poodle's comfort, we would suggest choosing a designated area that would hold her dog bed and toys. This can be a small gated off area or an indoor canine playpen. You would leave the entrance to this open when you are home (so your Poodle can enter into it when tired and wanting to take a nap) and you would close it at night, to keep her secure and in one spot while sleeping.
2) It is human instinct to want to check on a puppy all the time, just to 'make sure'. However, there is no reason to check on her as often as you do. Just the sound of you approaching may wake her up; dogs can hear so much better than we can. As you are tiptoeing up to her cage, most likely you think you're being super quiet but she hears you coming.
So, from her perspective, she is sleeping (or at least resting quietly), you come up to her. She believes that you are initiating interaction, so she in turn runs to your room. But wait! You really didn't intend for that... how is she to know? You put her back in her cage and now she's really confused. Is she supposed to sleep? Is she supposed to wait one more hour until you tiptoe back in? Is it a game...that if she plays along, at some point (about 5:30 AM as you mention) the 'game' ends when you finally take her out from the cage and she gets to be with you... sitting and taking a nap on your lap?
With your Poodle puppy being 4 months old, she can hold her urine for 4 hours. For this reason, it would be best to only check on her every 3 and 1/2 to 4 hours. It is important that Charlotte learns to control her bladder as she grows up. At 5 months, she should be able to hold her needs for 5 hours...at 6 months, 6 hours...at 7 months, 7 hours...And then no matter how old she is, 8 hours will be the maximum time that she will be able to hold her bladder.
Once you are only checking on her every 3 and 1/2 to 4 hours, she will sleep better. And there is no need for you to be waking up so often. Once you stop interrupting her resting/sleeping time, she will learn to hunker down and will sleep better through the night.