What does the Cinnamon in my coffee turn into?
Cinnamon is made from the ground bark of a variety of trees in the Cinnamomum genus. All of these barks contain starches, soluble fibers, and insoluble fibers to some degree.  

Lower grade cinnamons such as Cinnamomum cassia contain higher amounts of lignins, bassorins, pectins, and mucilages; accounting for almost 80% of the mass of the powdered cinnamon.  These compounds form gummy, mucus-like masses when hydrated or dissolved in water.  The mucilages in cinnamon can be dissolved in cold water, but it takes heat and time to fully hydrate the bassorins and other compounds which creates a gummy mass.  Higher quality cinnamon from different species may contain less mucilages or other compounds, but all powdered cinnamon will form a mass to some extent.  Herbs and spices such as cardamom, chicory, nutmeg and many others contain many similar compounds in various proportions that could also form gummy masses, but it would take research and/or experimentation to figure out the amounts needed.

Cinnamon flavors can be added to coffee without ending up with sludge in your cup by adding the cinnamon to the brewing chamber or by adding cinnamon extract or cinnamon essential oil to the brewed coffee.