"The children are not inside, they are outside walking by the bridge."
Translation:A gyerekek nem bent vannak, hanem kint sétálnak a hídnál.
"Hanem" is an important part of that "Nem X, hanem Y" construction in the Hungarian sentence. It can occasionally be omitted, but the sentence can feel "incomplete" without it. The second clause is in contrast with the first one, and "hanem" is the word that connects them. In the English sentence, "they (are)" provides that same kind of connection, by referring back to the subject of the first clause.
We can make it simpler by splitting the sentence in two.
"The children are not inside" - "A gyerekek nincsenek bent".
"They are outside walking by the bridge" - "Kint sétálnak a hídnál"
No "but", no "hanem", and even "nem (bent) vannak" can be changed to "nincsenek (bent)". Because the two sentences are not necessarily in contrast anymore. They are just two separate statements.
So, as such, we can reattach them, in their new forms:
"A gyerekek nincsenek bent, kint sétálnak a hídnál".
There you go. The contrasting structure is gone. And you can NOT place "hanem" back in this sentence whithout switching back to "nem bent vannak".