Is what the husband did by preventing his wife from traveling to visit her sick mother without a mahram correct, and what should he do regarding her request?

A wife must obey her husband in what is right and beware of displeasing him, for the Prophet, peace and blessings be upon him, said: "There are three whose prayers do not rise above their ears: ... and a woman whose husband goes to bed displeased with her." The husband must permit his wife to visit her mother who is ill with a sickness that is likely a death illness, provided her travel is with one of her mahrams or a trustworthy company. As for an ordinary illness, he should not permit her to travel unless it is with a mahram.