Question

"improving personal relationships" is a poorly phrased specific purpose for a speech because it is

Answer 1

I believe the answer is: written as a fragment rather than a full infinitive phrase.

The purpose of a speech should generally targeted to all audiences of the speech.  'Improving personal relationships' is something that could only be done to a handful of people and could only be done by communicating with those people directly, not by speaking to the crowd.