  We show that the difference between the topological 4-genus of a knot and the
minimal genus of a surface bounded by that knot that can be decomposed into a
smooth concordance followed by an algebraically simple locally flat surface can
be arbitrarily large. This extends work of Hedden-Livingston-Ruberman showing
that there are topologically slice knots which are not smoothly concordant to
any knot with trivial Alexander polynomial.
