What's the difference between the two words?
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The dictionary you linked to actually kind of clouds the view on a difference that is pretty obvious to every native speaker. "Fügen" is all about physically fitting something into a existing gap, slot or form. It is also used in an abstract sense. "Sich fügen" means "to comply". You fit yourself into the frame given by the law or whatever). It can or cannot imply addition. You can nicely grasp the word of you look at the meanings of "Fuge" (here)
"Hinzu" adds or strongly underlines the idea of "additional". You add something but it also fits nicely.
"fügen" is the basic or "simplex" verb without a prefix. It can be used in different ways, see an online dictionary. It is related with Fuge, a joiner's term and the technic of "fügen" is primarily to join wooden pieces in a way that the connection is very close and durable. Wooden boards of a floor are normally "gefugt" by groove and tongue.
The word family of "fügen" has sich fügen (figurative use); verfügen über etwas, zwei Teile zusammenfügen, etwas zwischen andere Teile einfügen.
You can say: einen Buchstaben in ein Wort einfügen, wenn man einen Schreibfehler gemacht hat.
PS This is a mere dictionary problem; when you say you can't solve the problem with the dictionary, you say you are not skilled enough to use either online or book dictionaries. Probably you don't know which dictionaries to use. I would be interested to know which dictionaries you normally use ( on line and bookform). I am sure one can give you recommendations in this matter. This is one of the first questions in language learning: Do I have or use the right tools as to dictionaries and grammars?