The pump injects air into the exhaust ports or manifold, and hot exhaust is prevented from entering the pump by the use of check valves. Pump failure is often a sign of a failure of the check valves. I don't know your model, but if intake is from the air cleaner, then these failed parts could allow hot exhaust gases unrestricted entry to the intake, which will cause running problems in some conditions. Just a thought.

Steve

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'91 MB 190E 2.3
'08 RAV4 Ltd 3.5
'83 Lazy Daze m'home 5.7