Most of the poor do not earn the minimum wage and most of those that do are not poor, so raising the minimum wage may not be the most efficient means of helping the poor.
If you want to help the poor give them money. The easiest way of doing that is giving everyone a minimum amount of money and taxing back that above the minimum. The reason this is done through wages rather than directly is to encourage productive work. The down side is it misses those unable to do so. No solution is without its problems. Providing money without a minimum wage would discourage work. A negative income tax would be more efficient in distribution but less efficient in collection. Encouraging lower productivity work would encourage employment but discourage innovation.
We could use more information on the poor that don't earn the minimum wage, and whether they are retired, disabled, unemployed/unemployable, employed at higher wages but seasonally or limited hours, students, single parents, large families, or experienced a capital loss. Just as low wage may be a poor measure, poor may be as well.
I can see three scenarios, 1) there are no gains only losses, those with lower productivity lose their jobs, 2) employers are labor short and forced to pay them more than their productivity, there are gains for workers but costs to employers whether passed on to customers or not, or 3) employers are labor short and invest to raise their productivity, gains all around.
I place little emphasis on scenario 2. I don't believe the employer will accept lower profits other than temporarily and if the employer can raise prices, they have raised their productivity. The argument against a higher minimum wage is not that it doesn't much help the poor but that it will mean more unemployment among the unskilled, fortunately that falls mostly on the non-poor. It will help the skilled somewhat, and innovation can help everyone more. Still, there is little evidence lower real wages (due to inflation) increase unskilled employment. Mostly, it just causes people to leave or not enter the workforce, but this is likely due to the minimum being below the market wage in substantial portions of the country.
One should equate wages to the marginal productivity of the work, not that of the worker. This is why raising the minimum wage can increase productivity; it can induce the investment necessary to do so. This is not without cost, and there are limits to what it can achieve, but this may be a better reason to support it than helping the poor.
A minimum wage can be useful if it leads to productivity boosting investment, or as a demarcation between welfare and work, but those are fairly limiting roles for it.