"Their income does not reflect the amount of work **that** they do" is correct. "That" in this sentence is a relative pronoun, referring back to something previously mentioned (the amount of work). However, in spoken English the "that" is often dropped and people would just say "Their income does not reflect the amount of work they do".
You could also say "Their income does not reflect **what** they do." Here you use the pronoun "what" because you aren't referring back to something that has already been said.