The draw of power upon keying up could be dropping too low on the radio and it resets. I would check the ground and power wire.
+1. It definitely sounds like inadequate current available when the transmitter fires up.
For the op, depending on distance between the radio and the battery, the hot and ground wires should be at least 14 ga if a 25 watt radio or 12 ga if a 50 watter. Distances longer than about 10' will require at least one wire size larger.
The hot lead should be direct to the hot terminal of the vehicle's battery. The ground lead should go to a known good factory chassis ground near the battery, but not the battery's negative terminal.
It's likely that there is also an ignition sense wire going from pin 10 on the back of the radio to an ignition controlled circuit. It's low current so it should be fused at 3 amp max and could be #16 or even #18 wire. You should also make sure that the ignition sense wire isn't wrapped around the antenna coax. RF can do strange things to 12vdc equipment.
Double, double check the connections at the battery and ground and at the radio. Any corrosion or oxidation should be removed and the metal should be clean and shiny. I like to place a very light coating of dielectric grease on all connections.
Also inspect the hot and ground wires from the battery to the radio. Make sure there is no damage or other obvious issues. I like to place my power wires in loom for extra protection when I do an install.
Get back to us on this and let us know if you found an issue and that fixed the problem.