If I have cooked chicken and need it to used by Sunday, can I make soup with it and the soup have an extended shelf life? Or do I need to throw out the soup on Monday?
You might be able to get a little bit of extra time out of the food if you’re boiling the soup long enough to pasteurize it.
But this doesn’t instant “reset the clock” on the ingredients, so I would still recommend consuming it within a few days of cooking.
The issue is that there are two factors here… there are microbes which can create toxins, and there are the toxins themselves. Pasteurization is to kill most of the microbes down to a sufficient level that they would need either re-colonize or build back up to a sufficient level to do most people harm.
But some of the toxins that they can produce requires even higher heat levels to destroy. So although you can repeatedly cook something, if it is repeated chilled and then recooked it may become unsafe. (There’s a question on here about ‘never ending soup’ that’s chilled each day, so unsafe for this reason: Never ending soup; is it actually safe?)
Note that small children, the elderly, and people with weakened immune systems are more susceptible, and so need a higher margin of safety.