Minimum wage means "All employees you hire must be paid at least $X/hour". Note that if this means that employers decide to fire all employees below that wage (or not hire them), the average amount of money going to poor people would be less than before. Maybe employers fire some, and raises the wage for others, and it's wash. Maybe they do raise the wage for everyone below the minimum, and poor people do end up earning more.
I'm not claiming knowledge of which of these cases will happen. My point is that we shouldn't simplify complex situations to simple descriptions and argue against strawman arguements.
Another way of putting it: if it really was the case that minimum wage means "paying poor people more", why don't we just set minimum wage to $20/hour? $30/hour? $50/hr? Surely that would give even more money to poor people, and poverty rates would be reduced even further! (Because there are tradeoffs involved, and we need to balance them carefully.)
If it's really the case that minimum wage means paying poor people more, surely we could just set the minimum wage at $30/hr and pay poor people even more.
I'm not even opposed to the minimum wage myself. I just don't think it's good to simplify a complex situation, and then use that to strawman someone's legitimate point (that it is not a given that minimum wages will reduces poverty rate).
[0] - Another example is that minimum wages vary across jurisdictions, and employers are able to arbitrage that fairly effectively for many jobs.
You might find this CBO report useful. https://www.cbo.gov/publication/55410