Poor adjective - Lacking money or material possessions.
Usage example: every year, we make up a basket of food at Thanksgiving for a poor family in the neighborhood
Short is a synonym for poor in impoverished topic. In some cases you can use "Short" instead an adjective "Poor", when it comes to topics like meagre.
Short adjective - Not coming up to an expected measure or meeting a particular need.
Usage example: regrettably, the art supplies are short this year, so you'll have to share
Both words in one sentence
- The Dog Bites Back: After a lifetime of being teased because he is short and poor, Judge Selah Lively uses his elevation to the bench in order to get back at everyone who tormented him.
- Then there's the elves, the rich/poor divide in the city—the short version is that in Dragon Age II, no one comes out smelling like roses.
- They aren't very good weapons: their range is quite short and their penetration is so poor that it takes near a dozen nails just to incapacitate one mook.
Source: Nail 'Em
Google Ngram Viewer shows how "poor" and "short" have occurred on timeline