You really need to replace the exhaust manifold and the doughnut gasket there first. This exh. leak will cause lots of problems. Notice I said exh. manifold, yes the whole manifold.

As far as that black plastic "tree" looking thing, it can't be stuck because it's just a piece of plastic. If it's broken, replace it and the o-ring that's under it. The humming noise is coming from the metal cylinder thing hanging below the "tree" in the black rubber sling. That is the idle air control valve.

Worry about the exh. leak first.

Pete

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If you can't afford to take your car to the dealer for maintenance, then sell it. Don't cry when the local shops can't deal with the technology anymore and you're tired of guessing at what parts to throw at it.