We report the calculation results of the effect of screening on weakly coupling Frohlich polaron binding energy of the ground state and polaron contribution to the mass for the lowest spin-down subband of the conduction band in a diluted magnetic semiconductor nanotube. The expressions for the energy of the polaron and the polaron contribution to the effective mass was obtained as a function of the magnetic field and the radius of the nanotube, taking into account the screening effect as well as virtual transitions from the ground state with n = 0 to the size-quantized subbands with <$En prime ~=~symbol С~1,~symbol С~2,...,~symbol С~10>. The performed numerical calculations demonstrate that the effect of screening decreases the polaronic binding energy and polaronic contribution to the effective mass about 35 and 25, respectively, for the value of the magnetic field equal 0,1 T. It has been established that the effect of screening reduces polaron binding energy of the ground state approximately by 30 - 15 and also reduces the polaron contribution to the effective mass by 25 - 10 with increase of the nanotube radius in the range about 10 - 40 nm.