@rachelbrookes first off, ***do not*** just toss what you have! We need to stop the ***insanity*** of wasting valuable resource by discarding products that are repairable. Repair will keep our hard-earned money in our pockets instead of further feeding the Profit-before-People (Planet) machinery.
The very first thing we need to know is the exact make and model for your TV. then unplug it (if you have not already done this) and remove the back cover. Now take a couple of really good , clear, large enough pictures of every circuit board you can see. Post those with your QUESTION. That will allow us to see what you see. Include the area with the black mark
Going by your description alone, this sounds very much like a blown component. From the noise you heard to the black mark on the case, it is most likely a MOSFET. Again, let's see what you got going on here.
Use this guide [guide|21499] to add the images.
***Repair is War on Entropy!***