[en] In a recent brief report [Kumar et al., Phys. Rev. C 89, 027601 (2014)] we proposed a method to include the effect of a large break up in fusion reactions with neutron and proton halo nuclei. A stronger enhancement on the total fusion cross section was found for the proton case. However, neither complete nor incomplete fusion was studied and only one target was considered. Here, we revisit the problem in order to address the importance of complete and incomplete fusion. The sensitivity of the cross section to the target is also explored. (paper)