Slight hand injury – does a hand wrap do the same job as a compression wrist brace?

Hi, I was training free and easy on bag and was doing left punches with kicks as an exercise, so I did not put on the right hand glove (yes super silly me, out of laziness).

I accidentally punched the bag with the right hand and the wrist is painful. I’m hoping it’s just inflammation or if ligament damage, just one that will heal with rest.

But I think I need some kind of wrist brace cause it hurts. Don’t really want to spend the money if it isn’t necessary – does wrapping your hand with handwraps actually do the same job as a compression brace?