귀찮게 하다 is not listed in the dictionary because it's a two word phrase, 귀찮게 + 하다, where 귀찮게 is a conjugational form.
It is constructed as 귀찮(다) (feel bothered, annoyed) + -게(a verb ending for "to make/effect") + 하다(do, make).
-게 turns verbs and adjectives to adverbs, so ~게 하다 is a causative phrase (슬프다 -> 슬프게 하다 - make someone sad).
So 귀찮게 하다 means "make someone feel annoyed", that is, to bother someone.
All three components are fully explained in Naver's Korean dictionary, but the English dictionary's -게 page is a bit lacking.
A better way to look up a phrase when you're not sure about the base form is to use the translator tools.
Both translate.naver.com and translate.google.com give correct translation of 귀찮게 하다.
Naver also offers the bonus feature of complete individual word definitions, for both sides of the translation.