I think that if you are so impoverished that you are relying on donations from random strangers then you are very grateful for any assistance you receive because that means you can provide for yourself and your child, I think these donations likely save their lives and they are greatly appreciated.
I think anyone would be grateful for any donations that they receive, but especially beggars with children are grateful for the donations they receive. They have more children to take care of with a hard financial situation so they are more appreciative when they do receive donations for their children.
Gratefulness is an emotion that requires cultivation, so it depends on the emotional state of the person who is getting donations. I don't think the life of a beggar leaves much room for contemplation even when they do have children. Their life is too hard to feel the sense of peace that comes with gratefulness, because they have to move on to the next person they can beg from in order to survive. It is wrong to me to donate something to a beggar and expect them to be grateful.
It is my opinion that beggars who have children are grateful for any donations that are made to them. With these donations they are more able to provide for their children, and hopefully themselves as well. I feel that they would most likely use the majority of any donation to purchase something for their children.
As a parent who has financial problems but not enough to the point that I have to beg, I would have to assume these people are very grateful for the donations they receive. As a parent, one of my main problems living in poverty is the fact that I can't provide for child to the degree I would if we had more money to spend.