Some people use an adapter to use their external antennas with their handhelds when they’re in a vehicle.
Noise on a mobile radio when moving can be caused by vibration exacerbating a poor connection somewhere in the feedline to the antenna or the antenna or mount, or by a bad power or ground connection leading to the radio (as well as other connections). Try hooking you external antenna up to the handheld and go for a drive. If the noise occurs then you have reason to suspect the feedline or antenna mount.
Otherwise inspect how you power your radio.
And maybe the radio itself is just affected by vibration. While operating stationary try shaking the radio to see if the problem occurs.
Finally, don’t discount picket fencing.
Good luck!