Cut away all of the burned area using your scalpel.
Use tweezers to insert a canvas patch through the opening you created by cutting away the leather. Position the patch so it lies flat beneath the leather.
Lift a corner of the leather up. Squirt a small amount of glue on the edges of the patch to affix the patch in position beneath the leather layer. If your glue doesn't come in a container with a thin, tapered applicator tip, spread the glue on your scalpel and use that to apply it. Wipe down the scalpel afterward.
Allow the glue to cure for 20 minutes.
Spread filler smoothly over the patch with your palette knife, creating a surface that is even with the leather. This may take up to four applications of fresh layers. Blow-dry each layer for three minutes before adding more filler.
Apply leather recoloring dye to the repaired area, using a cotton swab. The dye must match your couch precisely or the repair will be obvious. Most dye manufacturers can create custom color matches if you send them a leather swatch. You can also purchase a leather color touch-up kit that contains numerous colors, allowing you to blend your own customized dye.
Blow-dry the recolored area. If the color isn't quite saturated enough, apply more color and blow-dry again.
Apply leather conditioner to the entire couch with a sponge. This will increase the suppleness of the leather and prevent it from drying out. If the leather dries out, it can shrink and cause the repaired area to pucker.