Her husband gave her some money and she bought a house; do her children from her second husband inherit any of it?

Praise be to Allah.

Firstly: 

It is permissible for the husband to give his wife whatever he wants of his money, and that becomes her possession and is added to her property. 

If your father gave your mother some money and put it in her account, then this money became her property and she can dispose of it however she wants. And if she dies that will be part of her estate which will be inherited from her by all her heirs, including her children from her second husband. 

This is different from the case if your father had bequeathed anything to her after his death and he did not give her that money and put it in her possession whilst he was still alive, because that would be a bequest to an heir, which should not be executed except with the consent of all the heirs. If the heirs agree – and they are adults of sound mind – then it would become her property and all her heirs would inherit it from her, as stated above. 

Secondly: 

If your mother bought a house with her own money or with money that her husband had given to her, then this house is her property and is included with her estate if she dies, and it will be inherited from her by all her heirs. 

And Allah knows best.