How should a husband who wishes to perform Hajj act, when he rejects the idea of his wife staying with the children in Egypt temporarily to bear their responsibility, knowing that the costs of Hajj in Egypt are high?

It is the wife's right that her husband should not be absent from her for more than six months without a valid excuse. However, if there is an excuse, then there is no blame on him, as Al-Mardawi mentioned from Imam Ahmad. There is no objection to you staying alone after your wife travels for Hajj, as long as it does not lead to neglecting a duty. If it is possible for you to stay with your wife and children until you perform Hajj, that is better, and what you fear for your child can be avoided by taking the means of treatment and prevention. However, if their staying with you is difficult, then wait alone while being keen to convince your wife to be patient in staying with her children until she returns, and that she will be rewarded for that, and that the marital relationship is based on mercy, understanding, and considering the circumstances of the other.