Getting a cat to come to you is easy, you give it food and pets, and then stay calm when it's eating/enjoying.

Repeat until you've built trust, a sick or hurt cat will typically take longer to trust. Count on several days of repeating this without hitches (no sudden loud noise while you're doing it, etc).

Sometimes cats are desperate and everything turns up in a magical, calm way without bloodshed. But more commonly the next part is trickier, the cat will resist you picking it up (especially if hurt) or shutting it in.

Trick here is to be decisive and clear in your body language. Prepare a cat carry box with hard sides, feel free to prepare it with some textile smelling of you, be mindful that it will almost certainly be pissed on. Also bring a towel.

You will have to, in a calm manner, put the folded towel over the cat, and with it lift the cat into the carry. The towel is to trap legs so you won't be scratched, and if you manage to have it snugly around the cat, there's also a way to calm cats by gently pressing them down.

If you are unsure, slow, nervous, or hesitate in your movement, the cat will bolt. If you're too fast, loud, or big in movements, it will as well. Relax and do it in a deliberate motion.

If you release the cat from the carry, it will take considerable time to rebuild trust. Consider either going with it to the vet at once, or let it out in a quiet spare room with food, water, and litter box, and giving it a day or three to get accustomed to the room before letting it explore the rest of the place.

Don't get scratched by the cat, they can have some pretty nasty stuff on the paws, and some transmittable pathogens if anything draws blood or gets in your face/eyes.

Good luck!