In a Universal basic income system, will the government give extra money (or provide extra free service) to people who need more money?
Maybe, maybe not. Usually, advocates of universal (or more commonly, "unconditional") basic income favor using it to, ultimately, replace existing means-tested aid programs. Whether they would also have supplemental, non-means-tested, programs based on special needs (e.g., medical conditions) or whether they would address those some other way (or not see them as public responsibilities) varies considerably between which source of need you are discussing, and which UBI advocate you are discussing it with.