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We continue to study analytically the anomalous waves periodic Cauchy problem for the focusing Nonlinear Schrodinger equation. We present explicit analytic formulas describing the effect of small/gain on the recurrence of the anomalous waves in the case of one unstable mode. It turns out that very small loss or gain can essentially change the character and the statistics of these recurrence. In particular, we analytically explain the results of numeric simulations from the paper by O. Kimmoun, H.C. Hsu, H. Branger, M.S. Li, Y.Y. Chen, C. Kharif, M. Onorato, E.J.R. Kelleher, B. Kibler, N. Akhmediev, A. Chabchoub (2016).