I have a small circuit and I think I may have crossed something and
wrecked a transistor. I'm pretty much self teaching and want to see if
I have this figured out properly

I am fixing a small board that charges a battery and powers a device.
When the board is plugged in the output power works fine and the
device will function. The battery has charging voltage going to it
while plugged in.
When the board is not plugged in it will not power the device from
battery. I have traced the circuit. Between the battery and where the
power supply meet up there is only a resistor and a pnp transistor. If
I jump the collector and emitter on the PNP everything works as
expected and the device will receive power.

The PNP, only the C and E are soldered to anything, the Base is not
connected to anything. Why would this be ?

Doesn't electric charge flow on C and E if there is no flow on base ?

Did this transistor go bad ? if there is flow between C and E why
place this in a circuit ?

Just trying to figure out how this works and why only C and E of this
transistor are used ?

Thanks if anyone can shed some light on this for me.