H-holomorphic curves are solutions of a modified pseudoholomorphic curve equation involving a harmonic 1-form as perturbation term. Following Hofer et al. (Dyn Syst Ergod Theory 22(5):1451–1486, 2002), we establish an asymptotic behavior of a sequence of finite energy H-holomorphic cylinders with small dα-energies. Our results can be seen as a first step toward establishing the compactness of the moduli space of H-holomorphic curves, which in turn, due to the program initiated in Abbas et al. (Comment Math Helv 80:771–793, 2005), can be used for proving the generalized Weinstein conjecture.