Is it obligatory to fast in atonement for someone who swore to take a child to a certain place after prayer, and then the child changed his mind about going after the prayer ended?

The questioner is not guilty of breaking an oath (hanith) because her oath was made to honor the one sworn upon, but he did not accept it. Thus, the refusal came from his side. In such a case, some scholars maintain that no breaking of the oath occurs. Fasting is not sufficient for the expiation of an oath unless one is unable to feed ten needy persons or clothe them, as Allah Almighty says: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever does not find [anything] - then a fast of three days. That is the expiation for oaths when you have sworn. And guard your oaths. Thus does Allah make clear to you His verses that you may be grateful." {Al-Ma'idah: 89}.