This is a common problem. The radio doesn’t
draw a lot of current on receive, but it does
when you transmit. If the power supply can’t
provide the required current, or there are high
resistance connections in the cabling (such as
corroded contacts on fuses) then the voltage
drops below that required to run the radio, and
it may shut off or do any number of quirky
things. (In one case, the radio in my wife’s car
wouldn’t stop transmitting if she shut off the
engine while transmitting, due to the lower
voltage. That was fun...)
So the first step is to make sure that the power
supply can deliver the required current (which
will be somewhat lower at low power), then
check the voltage at the radio for any voltage
drop in the wiring.
Sometimes there are other issues with the
power supply, such as RF interfering with the
regulator (should be reduced using a dummy
load), or supplies that don’t respond quickly
enough to changes in current draw (which
may be OK for an FM transmitter, but cause
real problems with SSB or CW).
But an inadequate supply, or high resistance
in he connections, is the first thing to check.