From the age and make of the caravan, unless the water heater has be changed at some point it will have a Carver Cascade 2GE. The fact the pump is pulsing indicated it is a pressure switch controlled water pump.
The fact it does not do it when the cold water is used tells me the non return valve fitted to the pressure switch is functioning normally.
It is quite common for the pump to cycle a few times fairly quickly when some hot water has been drawn off, as the pressure needs to be rebuilt inside the hot tank., but this should not normally be more than one or two short bursts within a few seconds of closing the tap.
If its continuing for longer, then we have to suspect a couple of other possible causes.
Mel has already suggested the pressure switch may need to be adjusted. This can be necessary if the battery voltage changes, perhaps the difference between using an EHU when the battery is fully maintained, or being off grid and the battery voltage falls. - See the whale pressure switch instructions for setting.
The next thing I'd be suspicions of, is the water heaters non return valve (NRV), which may be jammed partially open. This can happen if the heater has not been fully drained and it be subject to a period storage during freezing weather. If ice has formed in the NRV, it can dislodge and jam the valves o ring seal. The classic symptom you will get is when opening a hot tap you'll get the normal flow of water for a few seconds then it diminishes to a bit more than a dribble. You may hear the pump continually turning on and off as the hot water is delivered slowly. The NRV is effectively restricting the water flow into the heater, and as the heater works on a displacement process, you can only get out what you put in. Depending on the pipe work, you can also sometimes get a pulse of warm water emerging from the cold taps. If this is the case the NRV needs to be replaced.
Changing the NRV in an old heater can be troublesome as they have a habit of breaking off as you try to remove them. It maybe best to refer a replacement to a skilled fitter.