Does the following passage contain a sentence fragment error?

To the kitchen for a towel and another cup of coffee to replace the one he spilled.

No, the passage does not contain a sentence fragment error.

Yes, the following passage contains a sentence fragment error. The phrase "To the kitchen for a towel and another cup of coffee to replace the one he spilled" is a dependent clause and cannot stand alone as a complete sentence. To correct the fragment, it could be revised as: "He went to the kitchen for a towel and another cup of coffee to replace the one he spilled."