Perhaps, but not so much by making the poorer wealthier, but by making the wealthier poorer. When the wealthy have access to new money they tend to buy assets, which provides a bigger more stable boost to their wealth than it would had they used the money on consumption as it would be used in your example.G, serious question, you said when more money is printed it only makes though who it initially goes to wealthier.
does that mean if 'new' money is given to the poor, or to fund services they use- such as public hospitals or schools, printing more money will help reduce inequality?
The benefit to the first user of the new money is a short term benefit, they too are hurt by the pernicious effect the money printing has on the production structure and on prices (in the longer term). The poor are the most vulnerable to increases in prices for consumer goods as a much higher portion of their income goes to it.
Why the fixation on inequality? Surely what is of more importance is generally well-being and access to resources to alleviate uneasiness? The poor Australian today has riches far beyond the kings of antiquity.