Translation:They really like white and black bread.
Someone else has mentioned marble rye, as far as one bread that is both, but I've heard of Russian black bread, which I think probably has rye in it.
(...now Boston brown bread was a disappointment for me--although I might still find a recipe I like--because I somehow always imagined it as a child as a sort of cross between homemade yeast bread and whole wheat bread or perhaps like steakhouse bread, but it actually contains a fair bit of molasses and corn and rye)
"They like very much white and black bread" means that the adverb "very much" modifies the adjective "white and black", so the result is that "very much white and black" modifies "bread" instead of the verb "like". Thus it sounds wrong to native speakers.
This is why we generally put intensifiers (very, much, really, extremely, etc) before whatever they modify.
I let my cursor hover over "tre" and Duolingo translated it "very." So I translated the sentence, "they like very much white and black bread," and Duolingo marked it wrong, and corrected it as "they like so much white and black bread," which is not even idiomatic English. Sad!
1.It is literal translation from Russian. In Russia rye bread called "black" and wheat bread called "white". Not usual definition in America. 2.About the sentence - I translated "they like very much..." Wrong, the Duo corrected "they so like..." When the sentence popped up second time I translated "...so like..." Wrong again! Duo corrected "...quite like". So I did "really". Passed!
"They very much like white and black bread" is accepted. I expect "They like white and black bread very much" would also be accepted. However, your version sounds as if it means something different - as though you're talking about the amount of white and black bread, in other words, "They like a lot of white and black bread". Perhaps that is why it's considered wrong.