These "card Slots" are not designed to "Come Apart".
It does depend on what is in the slot.
For simple cleaning, use a quick blast of compressed air, which you can get free from your local garage tyre pump gun, or compressed dry air from electronic/photographic stores in a small pressurised can.
If you have 'gum" of any sort down the slot, it can be frozen, by putting your player in a sealed plastic bag overnight, in the morning the gum will be solid, and lost its stickiness, thus can be eased out with a straightened out paper-clip.
If it is much more than the above, it gets more complex, and you need Isopropyl Alcohol (poisonous) in a small bottle from your local chemist/pharmacist.electronics supply store. Be aware that this liquid is also used in making certain banned drugs from simple materials, thus you or an adult may have to show Photo ID and sign with address etc, to obtain the Isopropyl Alcohol.
Using a thin lint-free cloth (A clean corner of a white handkerchief is good), moisten the corner of the cloth, and insert carefully, moving it along, backwards and forwards with a straightened paper-clip. Repeat as needed, turning the cloth as needed, until the cloth comes out "clean".
Try your SD card.
Let's know how you progress with solving this interesting problem....
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"The number of inventions increases faster than the need for them at the time" - SparkY
Are you sure this isn't an MMC card slot? MMC cards are thinner than SD cards, so an SD card won't fit an MMC slot. If this is the case, no amount of cleaning will help.
If the slot is the right size, and the card just won't go in all the way, canned air might help. If it goes in and seats, but doesn't make good contact, some kind of contact cleaner is indicated. General Cement has been making chemicals for electronics for many years. http://www.gcelectronics.com/ Take a look under "Chemicals". They sell contact cleaners that work quite well, but some of them are incompatible with some plastics. You might ask them for a recommendation. Nowadays they sell their excellent products only through distributors, so you will probably need to buy the stuff from a local electronics distributor, a dying breed these days. In the event, Radio Shack may be able to order it for you, but this is just a guess.
Ninety-nine per cent isopropanol is available under two dollars a pint at Safeway in California, is a good cleaner, and isn't especially toxic, but it trashes some plastics. Proceed with caution. Good luck.
And what everyone forgot is:- make sure that whatever the slot is in is switched off when you do any of these cleaning actions - battery removal, if possible, being a really good idea.....
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"What others say about you reveals more about them, than it does you." Anon.