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In association with the Blanford-Znajek mechanism for the extraction of rotational energy from a Kerr black hole, it is of interest to explore how much of magnetic flux can actually penetrate the horizon. For the completely uncharged Kerr hole case, it has been known that the magnetic flux gets entirely expelled when the hole is maximally rotating. In the mean time, it is known that a rotating hole immersed in a magnetic field, when it is surrounded by plasma, accretes a certain amount of electric charge. In this work, we show that, as a result of this accretion charge, small enough not to disturb the background geometry, the magnetic flux through this slightly charged Kerr black hole depends not only on the hole’s angular momentum but also on the hole’s charge such that it never vanishes for any value of the hole’s angular momentum.