Can I solder a wire between two leads on a PCB to make up for a trace that isn't wide enough?
Yes. Fixing production errors by using jumper wire is a time honored tradition by those blessed with hindsight.

For Power rails or low speed GPIO (Think Push Buttons inputs or LED outputs), a simple wire would work.  As a precaution, you could cut the trace, and carry the full load on the jumper you are adding. If the jumper fails, the load would try to go over the 10A trace, and can cause issues (i.e. fire).

High Speed Signaling or Protocols with tight tolerances, RF signal traces, Traces that require uninterrupted ground/vcc planes under them, matched impedance, etc, require significantly more thought into it.

Then again, some have wider tolerances than stated. There are many hobbyist projects/hacks that add high speed USB ports to a board with a CPU that supports it, using nothing but regular wires and protoboard, not to spec.