All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
If you marry the woman whom you mentioned in the question, her two children will become your stepchildren and their rights on you are the same as those towards a fellow Muslim. However, they deserve more care since they are your stepchildren. You will get a great reward by being kind to them and trying to nurture them on the obedience of Allaah and the obedience of His Prophet .
In fact, you are not obliged to spend on them but if you do so, you will be rewarded, and this is considered as being kind to your wife. Allaah Says (what means): {And live with them in kindness.}[Quran 4:19] We ask Allaah to enable you to do all that which is good.
As regards their rights on their father, then he is obliged to provide for them, feed them and cloth them if they are poor and do not have any money. Besides, what you mentioned that their mother made an agreement with their father that he would not provide for the two children, then it is permissible if this was in return for the father renouncing their fostering –as we clarified in Fatwa 88233 – or in return for their mother getting Khul' from him. Moreover, the father is not obliged to spend on his two children if the government sufficiently spends on them.
On the other hand, the father is obliged to be kind to his children. Allaah Says (what means): {And speak to people good [words].}[Quran 2:83] Of course, the children come in priority over other people in regard to speaking to them in good words and providing for them as long as they are young and poor. However, the father should not command his children to do things that would overburden them. If the father overburdens his children in a manner that they could not bear, then they are not obliged to obey him. Similarly, they are not obliged to obey him in things that entail the disobedience to Allaah. Moreover, the father is obliged to nurture his children on righteousness, as Allaah Says (what means): {O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones.}[Quran 66:6] For more benefit, please refer to Fataawa 84687 and 84618.