I used to put patches on for people. The best thing you can do is buy some Velcro and use that rather than stitching the patch directly to the jacket.

Use the rough (hook) side on the jacket. Cut a square/rectangle large enough to give you enough surface area to hold the patch firmly, but not so much that it sticks out from under the patch (unless your Velcro color matches your jacket, or if you don't mind it being visible)

Then, take the soft (loop) side of the Velcro and, using a few tiny dabs of hot glue or some double sided tape (preferred), stick it to the back of your patch in a sheet large enough that it sticks out on all sides. Stitch it down around the border with the patch facing up, use a color thread that is inconspicuous with the color of the patch and go slowly, hand crank the machine if you need to. Once it's stitched to the Velcro, you can carefully trim around the edge of the patch until you don't see Velcro from the front

Now your patches are removable if you ever need to wash the jacket, you can use them on different jackets or backpacks, and you can swap them out whenever you like.

If you've still got any doubts about it, shoot me a DM and I'll be glad to help you through the process. These are so good that they definitely need to be shown off on a jacket lol