Is it permissible to spend money sent for performing 'Umrah on the family to meet their needs after the increase in expenses and responsibilities?

How you dispose of money gifted to you depends on the intention of the giver. If the intention was for it to be spent on Umrah, then you must adhere to that. However, if they did not intend to specify Umrah, but rather their intention was merely a donation and a gift, and they mentioned Umrah as a way to present the money as a gift, then you may use it for any of your own needs. For instance, if someone gave him dirhams and said, "Buy yourself a turban with it," or "Use it to go to the bathhouse," then it becomes designated for that purpose, out of consideration for the giver's intent—if their intention was for him to cover his head with the turban and clean himself by going to the bathhouse. Otherwise, it is not designated, and he owns it or can dispose of it as he wishes. The intention of the donor is known through contextual clues, their words, or by directly asking them.