Light bulb sockets in ceiling fan damaged due to corrosion or arcing: Can clean to repair or must replace?

shut off power get some fine steel wool and jam it down there with a wood dowel and do your best to scrub out the sockets to get them as shiny as possible and make sure to remove the steel wood completely so there are no stray fibers, then install LED bulbs which do not draw that much power or generate as much heat and then cross your fingers.

Sockets of that type are readily available as lamp components, sold by competent hardware stores and lighting supply stores.

Don't continue running it - that's a mistake given that you know it's defective. Swap the socket for a good one.

Most likely the entire luminaire is an accessory that attaches to the fan hub, so it could come off and be serviced on a bench, or taken in to a lighting supply to be repaired. Note the ventilation holes at the top of the bulb housing; use them to see inside, or if necessary send down a screwdriver. Don't be surprised if the socket is screwed down onto a "hickey" (threaded hollow tube about 1/2" diameter, common in lamp construction).

If the bulb is getting excessively hot, it may be a cheap foreign job, or it may be an obsolete type called an "incandescent" which is basically an electric campfire that also makes some light. Getting modern bulbs should take care of that.

This happens all the time where arcing eats the solder on the bottom of the lamp. The lamp flickers or will not light.

An easy fix is with the power off use a screw driver or even a kitchen knife to lift the tab in the center of the socket. It is usually a brass tab that is folded. Don’t go crazy only a 1/10 of an inch is normally needed maybe slightly more to make good connections. I have done this 100’s of times and suggested this fix on this site as an accepted answer.

40w bulb should be hot it is using 40w. I have a soldering iron that is 40w and it gets up to 850f but is much smaller the bulb dissipates the heat over it’s entire surface.

If the brass flag breaks when you lift it the socket can be replaced the sizes are hard to mix up and lamps for a miniature lamp will not fit a standard fixture. The main difference is how they mount all but 2 of this style that I have replaced were screw on and they had a set screw to keep them from twisting.