I would expect a 4A MCB is too close to the 3.8A and would choose a 6A breaker
to avoid nuisance tripping use a type C or possibly type D breakers instead of the more usual type B
You might think about using a fuse instead of a breaker. An inline slow blow fuse could be a better choice than a breaker. They are smaller, and you can more readily make changes to the size or type of fuse. I would recommend starting with a 3 Amp Slow blow fuse. If that blows then change the fuse.
Also, it would be a good idea to measure the amount of current that is drawn under load. The numbers printed on the servo are probably worst case and, depending on your design, you may not want to put that much power into the servo. Only put as much power into the system as the weakest link. For example, you may be driving your load through a coupler with a tiny shaft diameter and secured with a small setscrew. If you load up the system and then apply maximum power you will surely spin the coupler on the shaft and the setscrew will carve a nice groove all the way around.
Finally consider voltage. Make sure that your circuit protection is rated at or above the maximum voltage.