Hi Nicholas!
Okay first it's important to try a memory card which you know for sure that it works 100%.
If not, buy a new one or test it in another PSP to see if it works.
If you did and it still doesn't work, then your memory card reader need an replacement... This is not easy and you need to solder it out and solder a new one in.
To give you the idea about how it works > https://www.youtube.com/watch?v=IKImM8FWiTk
Part: https://www.ebay.co.uk/itm/393992094813?chn=ps&_ul=GB&_trkparms=ispr%3D1&amdata=enc%3A1ybpdmCbFTrmUXUWsFhNBHA27&norover=1&mkevt=1&mkrid=710-169684-531343-4&mkcid=2&itemid=393992094813&targetid=325425753764&device=c&mktype=pla&googleloc=9193247&poi=&campaignid=20649934318&mkgroupid=153764631759&rlsatarget=pla-325425753764&abcId=&merchantid=231854446&gad_source=1&gclid=EAIaIQobChMIro_j0KKkiAMViYpoCR0Z1wzjEAQYAiABEgI0bfD_BwE
Please let us know how it goes! Hopefully you will get it fixed!!