Which takes precedence: buying a car for work and transportation with my mother, or performing Umrah with her, given the lack of sufficient funds to do both together?

If someone has money that they need to buy a car or other essential necessities, and they cannot find other money with which to perform Hajj, then Hajj is not obligatory for them. This applies even more so to 'Umrah. When choosing between 'Umrah and buying a car for someone who needs it, 'Umrah is recommended while buying a car is permissible. The general rule is to prioritize what is recommended, unless the permissible act assists in greater legitimate interests, such as maintaining kinship ties, spending on one's family, and becoming self-sufficient from others. 'Umrah is a means to open the doors of sustenance, eliminate poverty, and expiate sins.