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My best friend told his neighbor that he could not take care of the dogs this weekend.
My best friend told his neighbor that he could not take care of his dogs this weekend.
When he sat the plate on the glass countertop, it broke into a thousand pieces.
When he sat the plate on the glass countertop, the glass broke into a thousand pieces.
My best friend told his neighbor that he could not take care of the dogs this weekend.
My best friend told his neighbor that he could not take care of his<~~whose dogs? the friend's or the neighbor's? dogs this weekend.
When he set the plate on the glass countertop, it<~~vague reference broke into a thousand pieces.
When he set the plate on the glass countertop, the glass broke into a thousand pieces. This sentence has good references.
Both sentences have a small error in them.
In the first sentence, instead of saying "he could not take care of the dogs this weekend," it should be "he could not take care of his dogs this weekend." This clarifies that the dogs being referred to are the neighbor's dogs.
Similarly, in the second sentence, it should be "the glass broke into a thousand pieces" instead of just "it broke into a thousand pieces." This makes it clear that the glass countertop is the one that broke.
Corrected sentences:
"My best friend told his neighbor that he could not take care of his dogs this weekend."
"When he sat the plate on the glass countertop, the glass broke into a thousand pieces."