In a recent paper, Y. Hu has given a sufficient condition for the fundamental group of the $r$-th cyclic branched covering of $S^3$ along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of integers $r > 1$ for which the $r$-th cyclic branched covering of $S^3$ along the knot is left-orderable.
"On left-orderability and cyclic branched coverings." J. Math. Soc. Japan 67 (3) 1169 - 1178, July, 2015. https://doi.org/10.2969/jmsj/06731169