Does an existential sentence with "有" only carry a non-specific noun?
Yes, normally, in standard usage 


  zhūozishàng yǒu wǒde zuòyè 桌子上有我的作业


isn't supposed to be correct. You'll hear it a lot, though, especially if a different specific noun phrase is contrasted, such as


  … dàn méiyǒu nǐde zuòyè 但没有你的作业.


My Better Half, a native speaker, swears there's nothing wrong with 


  … yǒu wǒde zuòyè 有我的作业.


Addendum: The preferred way of dealing with stating the location of a specified noun (modified by a personal pronoun (wǒ 我, etc.) or a deictic pronoun (nà 那, etc.) is to make it the topic of the sentence, giving it greatest focus, as befits a specified noun. So 


  wǒde zuòyè zài zhūozishàng 我的作业在桌子上


etc.