Within the Landau-de Gennes phenomenological theory, we study the influence of an applied electric field with average strength E{a} on the position of a nematic line defect with topological charge M=+/-1/2 in a hybrid cell. We explore the biaxial structure of the defect core and we describe its expulsion from the cell upon increasing E{a}. We show that prior to the expulsion the defect core displays dramatic changes for strong enough surface anchorings. At a critical value of E{a}, the core broadens and merges into a surface layer with a large biaxiality. This transition corresponds to the reconstruction of the nematic order already observed in the bulk in response to an applied electric field. A similar order reconstruction could take place even in the absence of defects, but at a higher threshold.