The behavior of M=1 kink modes of reversed field pinch plasmas is analyzed in the presence of asymmetric external perturbations, especially field errors at toroidal shell gaps due to vertical field penetration. Perturbations in the vacuum and in the plasma are respectively expressed by using a class of modified Bessel functions and Bessel functions with M=1 modes, and amplitudes are determined by matching those on the perturbed boundary. It has been shown that the M=1 perturbation is amplified and its stability is deteriorated by the response of the plasma to the shell-gap field errors. Comparison with experimental observations of magnetic field fluctuations on the TPE-1RM15 Reversed Field Pinch is also made.