Suppose there is very little kinetic friction coefficient μ between the road and the carriage wheels. So we neglect that.
Let the mass of carriage is M (at time = 0) with no water collected in it. Let it move with a constant speed of 10 m/s at this instance. The force required to keep it moving at constant speed is zero, if there is no rain.
Now the mass increases at 50 gram/sec = 0.050 kg/sec. We neglect the impact of velocity of the rain in the vertical direction. The rain drops have no velocity in the horizontal direction.
So in 1 sec, 0.050 kg rain water has to be given a velocity of 10 m/sec. The momentum of 0.050 kg should change from 0 to 10 m/sec in one second, in horizontal direction.
force needed = rate of change of momentum = 0.050 * 10 m/sec / 1 sec
= 0.5 Newtons.
If this much force is given (extra) then cart will move at constant speed under constant rain. Otherwise, it will slowdown due to increase in mass.