Feeding problems?
Extractors and their spring tension are gauged during fitting.
If too much metal is removed during fitting (filing on the extractor's adjustment pad), the extractor hook will be too tight (since it reaches too deeply into the breech face). The extractor is ruined and you start over again fitting another one.
Once the extractor is correctly fitted (confirmed using the factory's GO/NO-GO bar gauge), then the spring tension is checked with a force dial gauge. If it's too heavy, feeding problems can occur. There's a recommended tension range for each caliber, and different springs are available that allow a tech, armorer or gunsmith to get the right tension for any particular gun.
While the bench checks using the gauges are necessary, it's also still necessary to make sure the gun actually works when live ammunition is used, so once all bench checks are good, then range testing is the final confirmation of proper functioning. (I've still had to make some minor adjustments to an occasional gun after live-fire testing.

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There are some potential situations where an extractor might require filing on the tail (the odd gun from a couple of older 3rd gen models), or a certain repair extractor spring set (nested or single) might be needed to get an older gun running properly (59XX), and the tension might range outside the recommended norm, especially in an occasional early production 3rd gen 9mm.
Also, S&W used to sell armorers a second extractor GO/NO-GO gauge, being a flag gauge, which checked the dimension between the back of the extractor hook and the breech face, but that gauge hasn't been used or available (or felt necessary) for several years.
Excessive dirt and fouling can also create some potential for feeding & extraction issues. Heavy fouling under the tail (like within the extractor spring and its hole) can make the extractor too "tight", and under the front (between the extractor's adjustment pad and the side of the slide's recessed slot) can make for extraction issues (not enough "bite", slipping off the case).
Chipped extractor? Time for a new extractor to be fitted. Probably time for a new extractor spring, too, if the chipping was caused by effects of normal long-term use, meaning work-hardening causing a brittle edge (and not user abuse, such as loading through the ejection port instead of from the magazine, damaging the extractor hook's edge).
Probably best to send it off to the factory, or an authorized warranty center, or take it to a gunsmith familiar with S&W pistols. Not a kitchen-table repair for someone not familiar with the repair, and who doesn't have the gauges (and parts

).
BTW, a damaged or worn out recoil spring can also lend itself to puzzling feeding problems.
As far as cleaning out the extractor recess?
If it's a 9mm/.40, there's a slot underneath the slide, to the right of the pickup rail, which opens into the bottom of the extractor slot. I've seen any number of folks allow excessive amounts of liquids to flow inside the extractor (around the outside of the extractor, in from the front and from underneath, at the machined slot). It ncan flow in but may not flow out quite as easily.
I use an air compressor (with safety glasses, and perhaps a mask and shop apron aren't a bad idea), in an open, well ventilated area. I put the tip of the nozzle up under the slot in the bottom of the extractor and direct air into the slot. You might be amazed at the amount of excessive liquid and black grunge that can ooze out around the side of the extractor.

(You also want to make sure you're not blowing moisture from the compressor up into the tight slide recess, too. Rust is bad for springs.)
The .45/10 slides don't have the machined slot on the bottom of the extractor recess, so they're harder to clean out. I still use a compressor, though, and press on them to pivot the extractor to "open" the front a bit.
I refrain from removing the extractor from the slide unless I'm replacing it or its spring. No need to create the potential for excessive wear & tear on the extractor pin holes (maybe wallowing them out), or damaging the slide (as it takes a healthy amount of force and a "starter" punch to break most solid extractor pins loose). I've seen the results of someone trying to remove an extractor in the wrong manner, using the wrong tools. The slide was ruined.
Any of this babble of interest to your situation?