Translation:There is a plate, cup, and spoon.
"There is" is the correct English translation for this sort of listing sentence using the definite article. "There are" would be appropriate if you are using the plural form of the items listed e.g. There are plates, cups, and spoons. Or a mixed list beginning with a plural e.g. There are plates, a cup, and a spoon. So there has to be plurals in used at the beginning of the list in order to use "are" in the sentence.
"There is a" is a valid sentence when you're making a list of singular objects. "There are" would be correct if there were multiple items.
An easy way to remember this is applying the "There is a" section to every object in the list. "There is a plate/cup/spoon" all make sense individually. "There are plate/cup/spoon" doesn't, so "are" would be incorrect.
But that's all really nitpicky. I have seen DuoLingo accept sentences with grammar that is far worse, so I'm suprised that it doesnt accept "are" here.
I thank martin.mk for the correction he made to your sentence, but regrettably, he provided no reasoning as to why. In general, whether the plural or singular existential form is used is dependent on the first element of a list in English. In this case, the first element is 'a plate', which is countable and singular, so 'there is' is generally accepted as correct English. The singular form also applies to situations where the first element is uncountable: "There is blood, bodies and destroyed vehicles everywhere!". The plural form is used when the first element is plural: "There are two apples, an orange and a bunch of bananas in the fruit bowl.". The rules can vary from region to region for collective nouns like 'people', in which case, both 'there is' and 'there are' could be acceptable. However, please do note, that, colloquially, the singular form is often used for cases where the first element is plural (but not the other way around).