Can I continue running the PC with damaged front USB pins on the motherboard?

Looks like you had a short in a power supply line near that header, which overheated things considerably. Eventually it got so hot that things separated enough that there was no longer a short. But something must have been limiting the current somewhat - otherwise the power supply would have shut down.

My concern would be that there's no reason (given that things there are just not mechanically stable) that ordinary thermal cycling couldn't cause things to move around enough that it wouldn't short out again, and maybe do more damage.

In other words, I'd replace it as soon as possible.