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I had a burn in a buckskin cloth bucket seat. I cut a small slit in the burn with a utility knife. I got a carpet sample of loop carpet
from a carpet store. They're glad to sell discontinued squares. I get them for cats to work out on also. Anyway, cut strands tie together with thread, hot glue and quickly push into slit with small screwdriver or awl. Now you have a ponytail you can trim with electric hair clippers. Maybe not the best or perfect fix, but better than the burn mark.
Check with either a used car lot or an auto upholstery shop to get a referral to a repair vendor. When we bought my daugher her car for college my wife had two cigarette burns fixed on the front seats - I believe it was $40 for both repairs. Three years and you still cannot tell where the repairs were done.