Is it permissible to go for Umrah with a sick uncle when the young man does not have money for expenses, despite his longing to visit the House of Allah and his fear of missing the opportunity?

The Umrah (definition) performed at someone else's expense is valid and sufficient if its conditions are met. It does not affect the validity of performing the obligation that one did not spend anything or that someone else bore most of its costs. Therefore, if your uncle covers your travel expenses for Umrah, do not refuse. Rather, perform it while seeking reward for the Umrah and for honoring your uncle, and you will attain both rewards.