"I like to eat Shanghai cuisine."
You don't need 很 here, but if you add 很 that is kind of emphasizing "I like it very much" in some context. 很 is an interesting adverb, sometimes it means "very" but sometimes it just connects words and has no special meaning.
Glad they accepted it without 很. The sentence did not say something like "really, a lot" so... why would I add 很?