Translation:Your child is under the table.
"Kid" is another word for "Child." __'s and "is" are interchangeable. For example: He's going-He is going; The book's over there-The book is over there
He means 桌子 is ambiguous, however general it is. Desk can also be 办公桌 (in an office) or 课桌 (in a classroom). Table can also be specifically 餐桌 (for dining), 台球桌 (for billiards), etc.
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"Your child is beneath the table" was counted as incorrect, but it shouldn't be
Can I put 是 before 在?
For example I want to say 'Monica Lewinsky is under Bill Clinton's table.' Can I write:
Monica Lewinsky 是 在 Bill Clinton 的 桌子下
Adding 是 makes this statement more confirmatory, though you don't need to alter the English translation. Such sentences are often paired with the contradiction “… 不是 …”. 是 can also work like (or be short for) 的确是, which means “indeed” but you'd better put a stress on 是 to mean “indeed”.
Perhaps this: “Adding 是 makes this statement more confirmatory”. (afterthought: too vague?)
What I wanted to say is that an extra 是 is often used to add some stress. Sometimes the stress is so small that we think it is a filler, but this doesn't mean we can freely include it.
是 can also be added to indicate certainty, functions as “indeed (的确)”. 的确是 is the more verbose version.
Example dialog (the words in parentheses can be omited at the cost of nuance):
你在房间里吗？对，我(是/的确/的确是)在房间里。Are you inside the room? Yes, I am (indeed) in the room.
那Amy呢？她(是)在厨房。Then where is Amy? She is in the kitchen. — stress for comparison
In (1) people can argue if 是 is simply used to emphasize that “I” am not at another location. Yeah, the ambiguity can exist but if 是 is stressed then it clearly means “indeed”.
What difference would there be if you wanted to say "a table" rather than "the table." I feel like I missed something, which i have been known to do just every now and then.
There can be no difference. To be explicit you can say 一张桌子, 某张桌子 and 某一张桌子. All of them can mean “one table”, “a certain table” or “some table”. But “某” is not quite suitable when the table is exactly pointed out by the speaker (just imagine pointing to it and say “he is under some table”).