Hi, welcome to Just Answer. My name is XXXXX XXXXX I look forward to helping you! There are several components that can cause this issue. If the water is warm, there is some heating going on, so I would look at connections, the heater itself and maybe a high limit thermostat. I suspect it to be a connection issue that is dropping voltage under a load. Let me know what you find.
They sound like good leads, but I don't understand them enough to actually do anything.
By 'conneections' do you mean to look for broken/disconnected wires?
How would I confirm if the issue is: connections, heater, thermostat?
What is a 'high limit thermostat'?