Hi. Welcome to JustAnswer.If the answer I provide for you is helpful, please click 'ACCEPT'. I won't be paid if you don't. You can use a good quality silicone caulking to seal the burnt area.
You need to scuff up the area around the burn with some 400 grit sand paper to ensure that the silicone gets a good grip on the plastic.
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The bath and tub caulking isnt the right kind. You need a good quality silicone sealant.
GE makes a good one that is used to seal drain gutters. Just ask the sales associate at Home Depot for it. It is a silvery grey looking sealant. Probably in the roofing department.