I need a glove to move with my puppet's arm

As David has said, the glove (think of it as a hand inside a glove) needs to be part of your character's arm. This is usually accomplished with a swap set. There would be a hand layer connected to the character's arm, and inside this layer would be all of the alternate appearances for that hand, such as an empty hand, a hand holding a phone, and your gloved hand. To make it work, when you are rigging your puppet, right-click on the hand layer and select "Create Swap Set." Once that is done, you can assign keyboard triggers to quickly switch from one version of the hand to another. This can also be used when you want the puppet to have different hand poses, such as waving, pointing, fist, etc.

You have to make the oven mitt as part of the puppet. Basically just make it one of the forms of the hand.  In the character below she had a basket of jelly beans that she would sometimes carry and in another scene used a remote control.  So I just have those set up with a trigger in Character Animator. Since they are part of the arm (wing, in this case) file when the arm moves the object moves too.

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