Is it permissible to eat food prepared by a female relative, which was purchased with money from a suspicious source, and to feed one's family from it?

There is no harm in eating from your relative's food, as long as you do not predominantly suspect that the source of the money is unlawful, because the default assumption for what is in a Muslim's possession is that it is theirs. If you predominantly suspect that the money is unlawful, then it is more prudent to abstain from it. You should also advise your relative to sever her unlawful relationship with that man.