I'm making a gluten free cake from a box and was wondering if there are ways to doctor the recipe to enhance flavors, much like we do with regular cake mixes.

No - being gluten-free, all the different types of flours and starches used in baking can be especially difficult balance to produce an acceptable product. Each starch and flour performs a specific function in the gluten-free mixture. Adding pre-gelatinized starch [usually corn or tapioca] in the form of pudding mix would have the effect of making the product denser rather than lightening the cake crumb.

The addition of pudding to a normal cake mixture reduces the protein level producing a lighter and more uniform crumb. Adding it to a gf cake mixture will have the opposite effect - there's no gluten reduce ..