I think that's a glitch. Collect the secret, and skip this for now. I'm sure someone at telltale could provide you with a save with the gum. I might be able to give you my save, which i think had all the items.
"The universe has to move forward. Pain and loss, they define us as much as happiness or love. Whether it's a world, or a relationship...everything has its time. And everything ends."-Elizabeth Sladen: RIP.