It is not clear what your question is asking.
Is it asking how much cargo must be moved to move the C of G from 38% MAC to 31% MAC?
Or is it asking where the C of G will be after when there is 6000 kg in Hold 1 and 4000 kg in Hold 4?
Initial condition.
MAC length = 4.6 meters, so 38% MAC is 4.6 x 0.38 = 1.748 meters aft of the MAC leading edge.
The MAC leading edge is 14 meters aft of the datum, so this is 14 + 1.748 = 15.748 meters aft of datum.
Total moment = total mass (190000 kg) x CofG position (15.748 m) = 2992120 kn m.
Final condition.
CofG is at 31% MAC which is 4.6 m x 0.31 = 1.426 meters aft of the MAC leading edge.
The MAC leading edge is 14 meters aft of the datum, so this is 14 + 1.426 = 15.426 meters aft of datum.
Total moment = total mass (190000 kg) x CofG position (15.426 m) = 2930940 kg m.
Calculation of cargo to be moved
Required moment change = new moment (2930940) – Initial moment (2992120) = -61180 kg m.
Moment change = cargo mass moved x distance moved.
Distance moved = new position – initial position
Distance moved = Hold 1 at (3.5 m) – (hold 4 at(20.39 m) = -16.89 meters.
Required moment change = -61180 kg m
Dividing required moment change (-61180 kg m) by distance cargo is moved (-16.89 m) = 3622 kg.
This means that 3622 kg of cargo must be moved from hold 4 to hold 1 to move the C of G from 38% MAC to 31% MAC.
But the final line of your statement of the questions states that “ Following the transfer operation, the new load distribution is: cargo 1: 6 000 kg; cargo 4: 4 000 kg”. If this is correct then only 3000 kg has been moved, so the new C of G will not be at 31% MAC.
Moving 3000 kg a distance of (-16.89 m) give a moment change of 3000 kg x (-16.89 m ) = -50670 kg m.
Adding this to the initial moment gives 2992120 kg m. – 50670 kg m = 2941450 kg m.
Dividing this by the total mass gives a new C of G position of 2941450 kg m / 190000 kg = 15.48 meters.
Subtracting the position of the MC leading edge gives 15.48 – 14 = 1.48 meters.
Dividing this by the MAC length then multiplying by 100% gives 1.48 m / 4.6 m = 0.322, which is 32.2% MAC.
So the new C of G position after moving 3000 kg of cargo is 32.2% MAC.