Welcome to the Dell Community  @JohnBlood 

I work on E6420's all the time and I have never seen that before???

The only thing I can think of for now is to make sure these drivers are up to date.

Intel Chipset Driver:

https://www.dell.com/support/home/us/en/04/drivers/driversdetails?driverid=t7wcw&oscode=w764&productcode=latitude-e6420

Dell Multi-Touch Touchpad Driver:

https://www.dell.com/support/home/us/en/04/drivers/driversdetails?driverid=vw486&oscode=w764&productcode=latitude-e6420

Intel HD, HD 2000/2500/3000/4000 Graphics Driver:

https://www.dell.com/support/home/us/en/04/drivers/driversdetails?driverid=xgpwm&oscode=w764&productcode=latitude-e6420

I wish I had more to offer.

Best regards,

U2

I think this is being caused by the computer switching from the brightness setting for when power is plugged-in to the one for battery power. I'm guessing that the laptop has a bad power connector because I've used this same power cord on other laptops with any issue. 

What do you think?

I haven't had a chance to update the drivers you suggested. Will do that tonight.

@JohnBlood 

I have seen where the laptop with a bad or under wattage AC will switch between AC charging and running on battery and the display will change in brightness or just give a flash when it switches.

What is the wattage of the AC adapter you have.

If it does not have the wattage on it just find the Volts and Amps.

Volts x Amps = Watts

You can look on the bottom of the laptop or under the battery for the Volts and Amps that the laptop requires.

The AC can be higher but not lower than what the laptop requires.

Regards,

U2 

I have a Targus charger outputing19.5V 4.61A

@JohnBlood 

You need to try an original OEM Dell AC adapter!!!

Do you get an "Alert" when you power-on???

Regards,

U2

@JohnBlood 

In the BIOS under Post Behavior:

Make sure "Adapter Warnings" is "Enabled"

Regards,

U2

No.