Your example sentence is more complicated than your description, because the phrase "frequently appearing" is modifying the noun phrase "election terms", so it is an [ADVERB + VERB] acting as an adjective. I'm not sure what to call that—I don't think it's an adjective phrase because it doesn't contain an adjective, but maybe attributive adjectival phrase would do.
In any case, the true ADVERB in your example is most. To ask the question posed in your title, we can substitute the word confusing for the two words frequently occurring, which gives us a true ADVERB + ADJECTIVE + NOUN set:
one of the most confusing election terms is the "underdog effect."
If we want to rewrite this using "that", we would say
one of the election terms that is most confusing is the "underdog effect."
which is NOUN + that is + ADVERB + ADJECTIVE
Going back to your original example, the adjectival phrase consisting of [ADVERB + VERB] needs a bit more inversion to be completely idiomatic. So
one of the most frequently appearing election terms is the "underdog effect."
could possibly be written as
one of the election terms that is most-frequently-appearing is the "underdog effect."
However, it would sound most correct to say
one of the election terms that is appearing most frequently is the "underdog effect."
Thus ADVERB + [ADVERB + VERB] + NOUN becomes NOUN + that is + VERB + ADVERB + ADVERB.