Eyelet replacement trace repair?
First, it is strongly recommended to clean the burned area using 99% isopropyl alcohol (IPA), and remove the burn mark as much as possible before attempting to fix the board, to prevent corrosive residue. 



Some people would also apply fresh UV-curable solder mask liquid to cover the damaged parts of the board to protect it from additional damages, but it's often not necessary. 



After cleaning, solder a jumper wire to reconnect the broken trace. Example: Pad 1 and Pad 2 are connected by Trace 1, which is broken. You can install a new component and solder it as usual, but then solder an additional wire from Pad 1 and Pad 2 to complete the circuit. Usually, you can follow the trace visually. But still, it would be the best if you have a schematic, or a known-good board, and/or an understanding of the circuitry, especially for multilayer boards, so you can check where the trace is going to.

I find AWG-30 wirewrap wire is useful for these repairs, just make sure to get a wirestripper.

 

For smaller components/boards, you may need to use a "0.1mm PCB Link Jumper Wire". These wires are easy to break, but you don't need to remove the insulation before soldering, the soldering iron will melt the insulation.



After soldering, reclean the board again to remove any flux residue, using 99% isopropyl alcohol (IPA).

You may want to glue the wire to secure it in-place. Hot-melt glue is not the best solution, but works reasonably well (make sure no other components is under the glue).