What does 'spent' mean here? I think the sentence would make sense without it.
After about 40 hours a week fatigue sets in, provoking mistakes.
Any extra hours spent are needed to clear up the mess: reversing poor decisions, soothing ruffled feathers.
You need the word "spent" because that explains what "any hours" means, ie, "any extra hours that are spent". The sentence would be nonsensical without it.
"Spent" means used, in the sense of "spend time".