"Die Kinder möchten lernen."

Translation: The children would like to learn.

[khanhu]

"The children would like to study" is rejected! Presumably duo doesn't believe that children ever like to study

[BBBThunda]

Would "The children like learning." also be acceptable? (Marked wrong 6/28/18)

[stepintime]

No, that would be "Die Kinder mögen es, zu lernen" (literally: "The children like to learn"; this sounds very uncommon, though), or much more commonly, "Die Kinder lernen gern(e)".
"Die Kinder möchten lernen" doesn't mean they enjoy it, but they desire it.

"Die Kinder mögen Lernen" feels awkward, but for other activities this is not an uncommon phrasing (mostly when talking about hobbies, otherwise it's preferably "Ich schwimme gern(e)"): "Ich mag Schwimmen" = "I like swimming"; the verb is turned into a noun here ("das Schwimmen"). (If it's not capitalised, "Ich mag schwimmen" would mean "I would like to swim" in informal wording, or it could mean that someone forgot to capitalise it and actually means "I like swimming".)