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TNR is only used for neutering and most of the staff may not have the skills to treat illness or injury. I wouldn't just take him in because they will likely turn him away.
Is he walking on the injured leg? Limping? Is there an open would or swelling that you can see?
I assume he is strictly outdoors and is around other cats. This being the case he may actually have a bite wound or an abscess if he is only limping but is still able to bear weight on the leg.
To answer your last question about ability to heal it all depends on the actual injury but even fractures in cats tend to heal quiet well with limited deformity and retaining use of the affected limb. Again, I can't put 100% into the ability to heal but I will say that cats are amazing creatures who tend to fair well against the elements and heal better than most species without intervention.
There isn't any wound. I checked
the leg is completely in the air
When I brought this guy in for TNR he had a terrible leg wound from a fight. It was very infected and he was sure to die. They took care of the wound and it has healed beautifully.
Right but they have already stated that they do not want to have him come in so I would try to avoid doing that if you can as they may choose to turn you away. Of course if it is your last resort and you cannot get him into a regular vet then you can give it a try. Of course you wouldn't be bringing him in for TNR this time so they could still turn him down. How about contacting your local shelters and see if they can recommend a low income vet if finances are a concern.