We numerically solve the Schrodinger equation, using a more general exponential cosine screened Coulomb (MGECSC) potential with an electric field, in order to investigate the screening and weak external electric field effects on the hydrogen atom in plasmas. The MGECSC potential is examined for four different cases, corresponding to different screening parameters of the potential and the external electric field. The influences of the different screening parameters and the weak external electric field on the energy eigenvalues are determined by solving the corresponding equations using the asymptotic iteration method (AIM). It is found that the corresponding energy values shift when a weak external electric field is applied to the hydrogen atom in a plasma. This study shows that a more general exponential cosine screened Coulomb potential allows the influence of an applied, weak, external electric field on the hydrogen atom to be investigated in detail, for both Debye and quantum plasmas simultaneously. This suggests that such a potential would be useful in modeling similar effects in other applications of plasma physics, and that AIM is an appropriate method for solving the Schrodinger equation, the solution of which becomes more complex due to the use of the MGECSC potential with an applied external electric field. (C) 2014 AIP Publishing LLC.