If more throttle is applied after engaging the reverse thrust, would it shorten the braking distance?
Broadly speaking, yes.
The standard method of operating reverse thrust on jet aircraft is to use the reverser levers, which are actually thrust levers as well. As you pull the reverser levers up, you increase engine thrust. Many aircraft types will have prohibited areas between idle reverse and full reverse thrust which should be avoided (for things such as blade flutter). Importantly, reverse thrust is not considered when evaluating aircraft landing performance - it largely used to reduce wear and tear on the aircraft for reasons of economy of operation, usually idle reverse thrust is ample for this purpose. As you suggest, in an emergency full reverse thrust can be used to reduce or prevent a runway overrun.
With propeller aircraft, turbine or piston, you may enter the 'beta' mode (if equipped), creating reverse thrust and selecting appropriate engine power as desired, though as with jets, there may also be prohibited engine speeds which may be passed through on the way to a higher speed, but not left at that level.