Adduce, advance, allege, cite may be used interchangeably in the meaning to bring forward by way of explanation, proof, illustration, or demonstration; however, they usually are clearly distinguishable in their implications and in their idiomatic associations.
One adduces facts, evidence, instances, passages, reasons, arguments when one presents these in support of a contention.
One advances something (as a theory, a proposal, a claim, an argument) that is in itself contentious when one presents it for acceptance or consideration.
Allege may indicate a bringing forward or stating as if needing no proof. It may on the other hand stress doubt about an assertion or convey a warning about or a disclaimer of responsibility for the truth of matter under discussion.
Its participial adjective alleged , especially, often serves as a disclaimer of responsibility for the assertion.
One cites only something concrete and specific (as a passage from a book or a definite instance) when one adduces it in support of a contention; one cites by quoting a passage to give an authority; one cites an instance that serves as a precedent or illustration; one cites definite facts in support of something (as a claim or proposal) advanced.