Lost Cat in Terrible Condition: Help Reunite!

This poor kitty came to my front porch in Hackham West where I leave out food for the local strays. I'm fairly sure it's a lost pet because it has a blue collar, but it's in terrible condition with fur missing and large areas are heavily matted. It's very dirty with leaves and twigs stuck in its coat. It may have either damage to its mouth or an infection as it has trouble eating. It will rub on my leg if I sit outside while it eats, but wont let me pick it up so I'm not able to catch it. Unsure of its sex. I'm hoping to earn its trust so I'm able to catch it and take it to a Vet.