Thank you for using Just Answer. My name is Rebecca. I am a veterinarian and will do my best to help.
The shiny part of the stool you are seeing in your cat is probably mucus; the lining of the colon can produce a lot of mucus especially if there is colitis (inflammation of the lining of the colon). The stools being mushy, and reddish are also signs of colitis.
Cats with colitis sometimes have to defecate urgently, and miss the litter box.
Colitis can be caused by many things: eating something irritating, a diet change, IBD (inflammatory bowel disease), parasites, infection, cancer. Trying a bland diet (boiled chicken with a little bit of rice, Royal Canin GI diet, etc.) may help. Probiotics (beneficial bacteria supplements) like Fortiflora or Proviable (available without prescription) may help.
If this has been going on for awhile, and if probiotics and a diet change do not help, she may need to be examined by your veterinarian and get prescription medication, or be tests for infection or parasites.
Let me know if this is helpful and what other questions I can answer.