Neighbours Cat is hurt and won't leave our house

Hey all,

we moved in a new apartment 3 months ago, and since the first day a cat started coming. We don't feed her, but she stays a lot of time at our place. We don't know to whom she belongs because there are a lot of apartments near by and we don't want to hang out our mobile numbers for all neighbours to see. Specially because we got some weird neighbours... she also doesn't wear a collar so we can't stick a note on it. So we just accustomed to it, and we love cats so there is no problem with that. At night we try to get her outside which normally succeeds investing some time. She doesn't like being touched so I won't do it. Problem is, today she came to our place as usual but she's hurt at her back. Not bad after my assumption just a little wound from a fight with some other cat I believe. She just went straight to her usual corner but no refuses to move out for the night. I had cats before and know they tend to retreat if they don't feel well. I am now sleeping in the living room in case just wants to get out at some point. But it doesn't seem to be the case.

What would you do in this situation?

We live in a very urban area so it's (99%) not a stray cat.